philosophy of physics

Causation in Classical Mechanics where P indic ates a Cauc hy princ iple value integral. We now assume that the integral along the semi-c irc le in the upper part of the c omplex plane vanishes as R goes to infinity,3 6 and we take that limit. We end up with (58) From here, we think in terms of the real and imaginary parts of ĝ(ω) = ĝr(ω) + iĝi(ω): Sinc e the sec ond summand in the last line is real, ĝr(ω) c an be equated to it. The first summand is imaginary so iĝi(′) c an be equated to it. One c onc ludes that the real and imaginary parts of ĝ(′) are Hilbert transform pairs. That is, (60) (61) These relations are known as dispersion relations (or Kramer's-Kronig relations). In the theory of waves in a dispersive medium, among the relevant func tions of a c omplex variable is the relative permittivity (or dielec tric constant), e, whose imaginary part gives the absorptive properties of the medium while its real part gives the dispersive properties. Sinc e the major driving engine of this derivation of dispersion relations is c ausality, it is typic al to c laim that c ausality implies suc h dispersion relations. But, of c ourse, other assumptions went into the derivation as well. In partic ular, it was assumed that ĝ(ω′) had no poles inside or on the c ontour of integration. A simple illustration of a system for whic h suc h dispersion relations hold is that of the damped osc illator. In this case, the real and imaginary parts of (16) are as follows: (62) and (63) Those are, in fact, Hilbert transform pairs. After this ground-setting, I want to suggest (following to some degree Norton (2009)) that, at a minimum, it is not c lear that c ausality is a fundamental princ iple that plays the role in dispersion theory that Frisc h believes. Both Norton and Frisc h foc us largely on the approac h in Jac kson's standard text. Part of what is at issue is whether Jac kson invokes c ausality in his derivation of dispersion relations. In the derivation of Kramers-Kronig relations, Jac kson starts (as is typic al) with an equation like (Jac kson 1975, 307) (64) where D is the displac ement desc ribing c harges in materials and E is the elec tric field. At this point, Frisc h sees “c ausality” as being invoked so as to limit the values of E that matter to D. He c laims, The next step in the derivation is to impose an additional c onstraint that is generally identified as a Page 17 of 24

Causation in Classical Mechanics c ausality c ondition. The c ondition is, as John Toll ([1956]) puts it, ‘no output c an oc c ur before the input.’ More precisely, we demand that the output field at time t is fully determined by the input field at all times prior to t. (Frisc h 2009a, 463). Frisc h sees the need for the c ausality c ondition as arising from a larger part of the interpretation of these equations: We do not take them just as func tional dependenc ies pac e Russell. Rather “we interpret them c ausally”; we interpret E as the c ause of D. This is supposed to explain why Jac kson follows (64) with (65) However, as Norton notes, this is not ac tually how Jac kson's derivation goes. Rather, Jac kson starts out assuming that g(τ) is the inverse Fourier transform of ε(ω) − 1 (i.e., of the relative permittivity minus 1). He had previously derived that ε(ω) − 1 is ωp2 . From there, Jac kson then derives that g(τ) is a retarded Green's func tion as we ω20−ω2−iγω did above for the damped harmonic osc illator. In fac t, one c an see that the equation for ε(ω) − 1 is relevantly similar to the Fourier transform of the damped osc illator Green's func tion. Jac kson never postulates “c ausality” in this derivation. Of c ourse, some assumptions went into the derivation of ε(ω) − 1. Jac kson derives the formula for it from an assumption about a “phenomenologic al damping forc e” ac ting on the elec trons of the medium in whic h elec tromagnetic waves are propagating (Jac kson 1975, 285). But, Jac kson has not c onc luded here that the Green's func tion must be retarded bec ause E c auses D. Rather, that the Green's func tion is retarded follows from a (partially phenomenologic al) model of the medium. So, it does not seem like the “c ausal interpretation” of the relation between D and E plays any role in Jac kson's ac tual derivation. Of c ourse, as (Frisc h 2009b) notes, Jac kson assumes a model for the medium that is not time-reversal invariant (sinc e it inc ludes damping). However, this is not the same as invoking c ausality as a restric tion on the model. That is something that Jac kson does not (initially) do. Rather, Jac kson adds a phenomenologic al damping term. So, he invokes the c ondition noted above in the disc ussion of the damped osc illator in sec tion 1.1.1: If there is phenomenologic al damping, add damping to the model. Onc e one has done that, c ausality (for that model) follows and is not needed as an independent c ondition. The appearanc e of phenomenologic al equations that are not time-reversal invariant is, of c ourse, something about whic h muc h has be written that c annot even be approac hed with any rigor in an essay of this sc ope.3 7 Moreover, many investigations into the sourc e of irreversibility take one outside of c lassic al physic s to arenas of physic s that are more appropriate to the mic rosc opic and also to grand c osmologic al c onsiderations. As suc h, in an essay of this sc ope, I c an only note that there is c ontroversy over what, exac tly, the sourc e of the irreversibility in Jac kson's model is. Leaving Jac kson's ac tual derivation aside, Frisc h c laims that it is not so muc h that c ausality is invoked in any partic ular derivation where a partic ular model of a material has been given. Rather, c ausality is invoked in some more general sense in the derivation of Kramers-Kronig relations. He c laims, Of c ourse, onc e we have spec ified a partic ular model for the dielec tric c onstant ε, the c ausality c ondition provides no additional c onstraint on that partic ular model. Rather the c ondition provides a general c onstraint on any physic ally legitimate model of ε and as suc h has c ontent going beyond what is c ontained in any finite list of suc h models. (Frisc h 2009a, 467) As applied to derivations of dispersion relations, the idea seems to be that we are able to derive dispersion relations without any partic ular model in mind: “a derivation of the dispersion relations that begins with [the c ausality c ondition] allows us to ignore the details of the medium in question and its detailed interac tion with the field” (Frisch 2009a, 468) But, for some models, Hilbert transform dispersion relations do not follow. For metals, ε(ω) has a pole at ω = 0 (Landau and Lifshitz 1960, 260). So, one has to take a c ontour around this point in a derivation of Hilbert-transform-like relations as above. Bec ause of this, the sec ond of the dispersion relations is modified by an addition of 4πσ /ω, where σ is the c onduc tivity. So, the model matters somewhat to what one gets out of these derivations and c annot be c ompletely ignored. Moreover, we are left with little sense of the status of the c ausality princ iple that is invoked. Just as it is unc lear what the sourc e of the damping is in Jac kson's model, it is unc lear what the sourc e of the irreversibility that results Page 18 of 24

Causation in Classical Mechanics in c ausality in material media more generally is. Typic ally, appeal is made to “initial c onditions” involving a low entropy past. But, it is not c lear that that has anything direc tly to do with c ausality. Initial c onditions are just a spec ific ation of the state (or sometimes a range of possible states) of the universe at a time. As suc h, they will not be expec ted to say anything about what c auses what or even what c an c ause what. At a minimum, someone of Russell's bent would want to hear more about why it is suspec ted that c ausality has a fundamental status if it results from spec ial initial c onditions. On the other hand, Frisc h does not seem partic ularly wedded to the idea that c ausality is fundamental. His main point is that it plays an ineliminable role in mac rosc opic elec trodynamic s. He c laims,” Even if the asymmetric c ausal c onstraint were ultimately in some sense reduc ible, it remains part of a genuinely sc ientific theory and within c ertain c ontexts is explanatorily indispensable.” (Frisc h 2009b, 491). As I have desc ribed Russell's c laims above, this is c onsistent with Russell's view, sinc e I limit the c laim that c ausation is not found in physic s to the fundamental equations of physic s. To be fair, it is not c lear whether this limitation is found in Russell. But, that is a question of Russell sc holarship, not a c laim that will take us c loser to understanding the status of c ausality in classical physics. So, I will not pursue it further. As for whether invoking c ausality is indispensable, more would need to be said about its indispensability and the sc ope of it. For linear systems, one need not invoke a c laim that the system in question obeys c ausality to derive dispersion relations. One c ould invoke instead that the system is “passive,” where a system is passive if it absorbs energy but does not c reate it. One c an show that a system whic h is passive and linear is also c ausal in the sense of (54) (Zemanian 1965, 300–303; Nussenz veig 1972, 391–392). Passivity is a notion whose relation to c ausality is not partic ularly transparent antec edently. Zemanian, for example, c laims that the c onnec tion between passivity and c ausality is a “remarkable fac t” (Zemanian 1965, 302). Prima fac ie passivity is a distinc t property that c an be invoked in the derivation of dispersion relations within the linear theory. As suc h, in a c ertain sense c ausality is not indispensable within the linear theory that Frisc h disc usses sinc e one c an invoke passivity instead. 5. Clo sing Tho ughts It would be diffic ult to summariz e the outc ome of this essay with respec t to whether the imposition of c ausality in the intended sense is important. Of c ourse, spac e limitations render any suc h disc ussion grossly inc omplete anyway. It seems to me, however, that Russell's position (at least as I have rendered it) looks somewhat better than is sometimes suggested. We have, at least, failed to find any c lear sense in whic h the retarded Green's func tion for a system like the undamped wave equation is privileged. Moreover, we have seen where a radiation c ondition (often thought to impose c ausality) is needed and where it is not. This has given us some sense that c ausality is a derivative c ondition used to inc orporate c ertain initial c onditions in c ases where there has been no oc c asion to implement them. Moreover, we have seen that there are grounds to dismiss the Abraham-Lorentz equation, but most of them remain even if one did not have sc ruples about bac kward c ausation per se. And, it is not c lear how seriously one should take the worry that the equation involves a c ausality violation. Lastly, we have seen that the assumption of causality only gets one so far in the derivation of Hilbert-transform dispersion relations. Rather, assumptions as to the material c onstitution of the medium are needed as well, sinc e one needs to know whether the Fourier transform of the Green's func tion has poles along the real axis. But, when one has a model of the material medium suc h as Jac kson gives, c ausality c an be derived rather than postulated. Moreover, within the linear theory one c ould invoke something other than c ausality in the derivation of dispersion relations. As suc h, Russell might argue that c ausality is not suc h an ineliminable, fundamental princ iple in linear dispersion theory. Even if I have not c onvinc ed the reader that Russell's skeptic ism is more warranted than sometimes supposed, I will be content to have framed the issues in a useful way. References Barton, G. (1989). Elements of Green's functions and propagation: potentials, diffusion, and waves. Oxford: Oxford University Press. Born, M., and Wolf, E. (1999). Principles of optics. 7th ed. Cambridge: Cambridge University Press. Page 19 of 24

Causation in Classical Mechanics Butkov, E. (1968). Mathematical physics. Reading, MA: Addison-Wesley. Cartwright, N. (1983). How the laws of physics lie. Oxford: Clarendon Press. ——— (1989). Nature's capacities and their measurement. Oxford: Clarendon Press. Colton, D., and Kress, R. (1998). Inverse acoustic and electromagnetic scattering theory. 2nd ed., New York: Springer-Verlag. Comay, E. (1993). Remarks on the physic al meaning of the Lorentz -Dirac equation. foundations of Physics 23(8): 1121–1136. DuChateau, P., and Zac hman, D. (2002). Applied partial differential equations. New York: Dover Public ations. Earman, J. (1986). A primer on determinism. Dordrec ht: D. Reidel Publishing Company. ——— (1987). Loc ality, nonloc ality, and ac tion at a distanc e: A skeptic al review of some philosophic al dogmas. In Kargon, R., and Ac hinstein, P., editors, Kelvin's Baltimore lectures and modern theoretical physics, 449– 490. Cambridge, MA: MIT Press. ——— (1995). Bangs, crunches, whimpers and shrieks: Singularities and acausalities in relativistic spacetimes. New York: Oxford University Press. ——— (2007). Aspec ts of determinism in modern physic s. In Butterfield, J., and Earman, J., editors, Handbook of the philosophy of physics, Part A, 1369–1434. Amsterdam: North-Holland Press. Erber, T. (1961). The c lassic al theory of radiation reac tion. fortschritte der Physik 9: 343– 392. Fetter, A., and Walec ka, J. D. (1980). Theoretical mechanics of particles and continua. New York: Mc Graw-Hill, Inc . Feynman, R., Leighton, R., and Sands, M. (1989). The Feynman lectures on physics. Redwood City, CA: Addison- Wesley. Frisch, M. (2005). Inconsistency, asymmetry, and non-locality: A philosophical investigation of classical electrodynamics. Oxford: Oxford University Press. ——— (2009a). “The most sac red tenet”? Causal reasoning in physic s. British Journal for the Philosophy of Science 60(3): 459–474. ——— (2009b). Causality and dispersion: A reply to John Norton. British Journal for the Philosophy of Science 60(3): 487–495. Greiner, W. (1998). Classical electrodynamics. New York: Springer-Verlag. Griffel, D. H. (1981). Applied functional analysis. New York: Halsted Press. Griffiths, D. (1981). Introduction to electrodynamics. 2d ed., Englewood Cliffs, N.J: Prentic e Hall. Hawking, S., and Ellis, G. (1973). The large scale structure of space and time. Cambridge: Cambridge University Press. Hilgevoord, J. (1960). Dispersion relations and causal description. Amsterdam: North-Holland Publishing Company. Hitc hc oc k, C. (2007). What Russell got right. In Pric e and Corry (2007), Causation, physics, and the constitution of reality: Russell's republic revisited, 45–65. Oxford: Clarendon Press. Hume, D. ([1748] 1977). An enquiry concerning human understanding. Indianapolis: Hac kett Publishing Company. Jac kson, J. D. (1975). Classical electrodynamics. New York; John Wiley and Sons, Inc . Johnson, C. (1965). field and wave electrodynamics. New York: Mc Graw-Hill. Page 20 of 24

Causation in Classical Mechanics Landau, L., and Lifshitz , E. (1960). Electrodynamics of continuous media. Reading, MA: Addison-Wesley Publishing Company, Inc. Levine, H., Moniz , E., and Sharp, D. (1977). Motion of extended c harges in c lassic al elec trodynamic s. American Journal of Physics 45(1): 75–78. Maudlin, T. (2002). Quantum non-locality and relativity. Oxford: Blac kwell Publishers. Norton, J. (2006). The dome: A simple violation of determinism in Newtonian mec hanic s. Philosophy of Science Assoc. 20th Biennial Meeting (Vancouver): PSA 2006 Symposia. ——— (2007a). Causation as folk sc ienc e. In Pric e and Corry (2007), Causation, physics, and the constitution of reality: Russell's republic revisited, 11–44. Oxford: Clarendon Press. ——— (2007b). Do the c ausal princ iples of modern physic s c ontradic t c ausal anti-fundamentalism? In Mac hamer, P. K., and Wolters, G., editors, Thinking about causes: From Greek philosophy to modern physics, 222–234. Pittsburgh: University of Pittsburgh Press. ——— (2009). Is there an independent princ iple of c ausality in physic s? British Journal for the Philosophy of Science 60(3): 475–486. Nussenz veig, H. M. (1972). Causality and dispersion relations. New York: Ac ademic Press. Parrott, S. (1987). Relativistic electrodynamics and differential geometry. New York: Springer-Verlag. Parrott, S. (1993). Unphysic al and physic al(?) solutions of the Lorentz -Dirac equation. Foundations of Physics 23(8): 1093–1119. ——— (2005). Variant forms of Eliez er's theorem. arXiv:math-ph/0505042v1, 1–10. Poisson, E. (1999). An introduc tion to the Lorentz -Dirac equation. arXiv:gr-qc/9912045v1, 1–14. Price, H. and Corry, R., eds. (2007). Causation, physics, and the constitution of reality: Russell's republic revisited. Oxford: Clarendon Press. Richards, J. I. and Youn, K. K. (1990). Theory of distributions: A nontechnical introduction. Cambridge: Cambridge University Press. Rohrlich, F. (1965). Classical charged particles. Reading, MA: Addison-Wesley. Russell, B. (1981). On the notion of c ause. In Mysticism and Logic, c hapter 9, 132– 151. Totowa, NJ: Barnes and Noble Books. Saff, E. and Snider, A. D. (2003). fundamentals of complex analysis. 3rd ed., Englewood Cliffs, NJ: Prentic e-Hall. Snider, A. (2006). Partial differential equations: Sources and solutions. New York: Dover Public ations, Inc . Steiner, M. (1986). Events and c ausality. Journal of Philosophy 83(5): 249– 264. Stoker, J. J. (1956). On radiation c onditions. Communications on Pure and Applied Mathematics 9: 577– 595. ——— (1957). Water waves: The mathematical theory with applications. New York: Intersc ienc e Publishers, Inc . Tikhonov, A., and Samarskii, A. A. (1990). Equations of mathematical physics. New York: Dover Public ations, Inc . Vanderline, J. (2004). Classical electromagnetic theory. Dordrec ht: Kluwer Ac ademic Publishers. Wallac e, P. R. (1984). Mathematical analysis of physical problems. New York: Dover Public ations, Inc . Weigel, F. (1986). Introduction to path-integral methods in physics and polymer science. Singapore: World Scientific. Page 21 of 24

Causation in Classical Mechanics Wilc ox, C. (1959). Spheric al means and radiation c onditions. Archive for Rational Mechanics and Analysis 3(1): 133–148. Wilson, M. (1989). Critic al notic e: John Earman's A primer on determinism. Philosophy of Science 56: 502– 532. Zachmanoglou, E., and Thoe, D. W. (1986). Introduction to partial differential equations with applications. New York: Dover Public ations, Inc . Zemanian, A. (1965). Distribution theory and transform analysis. New York: Dover Public ations, Inc . Notes: (1) Three rec ent papers that are sympathetic to Russell are Norton (2007a), Norton (2007b), and Hitc hc oc k (2007). See, more generally, the papers in Pric e and Corry (2007). (2) Whether this is ac tually an ac c urate interpretation of Russell is an open question. It does, I believe, represent a view that has bec ome assoc iated with his name. (3) Russell never puts the point this way, but he does c laim, “In the motions of mutually gravitating bodies, there is nothing that c an be c alled a c ause, and nothing that c an be c alled an effec t; there is merely a formula” (Russell 1981, 141). (4) My formulation of this c laim owes to Norton sinc e this is how he often expresses his own skeptic ism about causal principles (Norton 2007a). (5) Among those who argue this via examination of physic al theory are Steiner (1986), Cartwright (1983, 1989), and Frisch (2005, 2009a). (6) For interesting disc ussion of the status of determinism in c lassic al physic s, see Earman (1986, 2007), Norton (2006, 2007a), and Wilson (1989). For disc ussion of restric tions on the veloc ity of propagation, see Earman (1987) and Maudlin (2002). Obviously, in a relativistic c ontext, if there is c ausation over spac elike separation, then there is bac kward c ausation in some frames of referenc e unless one “reinterprets” the direc tion of c ausation in suc h frames. I will limit myself, however, to c ases of bac kward c ausation that are within or on the light c one. (7) Mathias Frisc h prefers to frame the princ iple as “the c ause does not c ome after the effec t.” This way of stating the principle seems to have as its sole motivation the desire to maintain that nothing is amiss with equations like Newton's Sec ond Law, f =ma, even though the forc e does c ause the ac c eleration but does not c ome before it, sinc e they are simultaneous. However, bec ause I will not disc uss alleged c ases of simultaneous c ausation, I will stic k with the more standard wording, sinc e nothing will depend upon the differenc e. (8) For disc ussion of “c ausality restric tions” in c urved spac etimes brought about by the existenc e of c losed-c ausal curves in some models of General Relativity, see Hawking and Ellis (1973) and Earman (1995). (9) Frisc h has appealed to nearly all of these items from physic s in support of the importanc e of c ausality c onsiderations within physic s (Frisc h 2005, 2009a). (10) Frisc h, someone who thinks that c ausality does play a substantive role in theoriz ing, points to this passage among others (Frisc h 2009a). (11) Typic ally, a Green's func tion will be a “weak solution,” whic h means that it is not suffic iently differentiable to be a solution in the c lassic al sense. (12) Some of what follows requires the theory of distributions to be made rigorous. (13) For a proof of this using the theory of distributions, see Ric hards and Youn (1990, 67). (14) As will be c lear from the disc ussion of the damped osc illator to follow, the c ontours depic ted in figure 3.1 represent the Green's func tion only for t 〈 0. Page 22 of 24

Causation in Classical Mechanics (15) In fac t, again, there are more than two, sinc e other c ontours of integration result in linear c ombinations of those two. (16) For disc ussion of their plac ement, see Butkov (1968, 277). (17) This is not hard to see. If we think of ω as a c omplex variable, ωr + iω i, the e−iωtin the numerator of the inverse Fourier transform is e−it(ωr+iωi)  = e−iωrt+ωit  =  e−iωrt eωit. When t 〈 0 and ω i 〈 0, the second exponent is positive and will grow as ω i gets smaller in the limit. (18) It is not obvious that there c ould not be anti-damped materials. (19) If one envisions c ases where there is forc ing before t0, one still has a rationale for taking f(t) to be z ero before that time. Either the forc ing before that time has no effec t on the state at t0 or its effec t is already fully taken into ac c ount in xi(t). (20) This is just another way of writing the general solution to (1) that uses a different partic ular solution to it (Hilgevoord 1960, 10). (21) Failing suc h differentiability, one may turn to a generaliz ed wave equation, but I do not disc uss the details of that here. (22) The details and setup of this problem are from the exc ellent Snider (2006, 554), whic h should be c onsulted for additional insight. (23) One might wonder why one assumes this. The reason here is that (below) we are trying to solve the wave equation via the Helmholtz equation and it requires the elimination of c ertain solutions that are unbounded at spatial infinity Suc h solutions c learly will not represent the solution to the wave equation we are seeking here. (24) To make matters rather c onfusing and misleading, I know of some texts that refer to it as the wave equation in spite of the fac t that it is not identic al with the wave equation above. In part, that is bec ause it does not involve the time variable. In essenc e, that has been partially transformed away and then further eliminated by the imposition of initial conditions. (25) One may find the solution to these problems in Snider (2006, 555). (a2n6a)l oGgiovuesn itnhceo tmimineg-d seopluetniodne nisc ee− rψikrt .( tT)h =e seu−miω to, fa t hsopshee rtiwcaol liys say smtamnedtirnicg owuatvgeo.in Sge seo Bluatriotonn i s( 1ψ9X8(9x, )3  3=6)  feoirkrr .a The disc ussion of the various types of solutions. The Sommerfeld radiation c ondition rules out standing waves in addition to incoming waves. (27) This point of view is further elaborated along with proofs in Stoker (1956). (28) Wilcox is thinking in terms of what is known as an “exterior boundary-value problem”: Conditions are given on the boundary of some bounded volume suc h as a sphere and one solves for the state of the field external to that surface. (29) For a summary of issues surrounding (25), see Erber (1961). (30) See Norton (2006) for a more detailed disc ussion of various notions attac hed to “unphysic al.” (31) One will be able to find something to c omplain about in nearly every derivation of the equation. Disc ussion of how rigorous derivations of the equation really are is ongoing as are attempts to derive it more rigorously. But, it is not c lear why our ordinary c onc eption of c ausality ought to be thought to be more sec ure than the steps in any particular derivation. (32) Eliez er-type results have been reinforc ed in Parrott (1993) and Parrott (2005). Even some who disagree with Parrott's earlier analysis c onc ede that there are problems surrounding Eliez er's theorem that await resolution (Comay 1993, 1131–1132). Page 23 of 24

Causation in Classical Mechanics (33) For an exc hange about c ausality in dispersion theory, see Frisc h (2009a), Norton (2009), and Frisc h (2009b). Some of this exchange is reviewed below. (34) See Fetter and Walec ka (1980, 315– 6), and Greiner (1998, 396). The rough way to think of the reason for this is as follows: if the Fourier transform c onverges for real values of ω, it c onverges better for values in the upper part of the complex plane. The integral ((59)) now c ontains the term e−τωi, whic h keeps the func tion regular when ω i 〉 0. (35) Sinc e we later take the limit as R goes to infinity, the c ontour ultimately enc ompasses the entire real axis. Sinc e the frequenc y, ω, will be real, it will, thus, lie on the c ontour. (36) If this is not the c ase, one may use the “method of subtrac tion” to get around that but I do not disc uss that here. (37) Time-reversal invarianc e does not hold in fundamental physic s. However, the failure of time-reversal invari- ance in the decay of neutral K mesons is not thought to be responsible for the sort of damping that makes Jac kson's model viable nor for thermodynamic al behavior more generally. Sheld on R . Smith Sheldon R. Sm ith is Professor of Philosophy at UCLA. He has written articles on the philosophy of classical m echanics, the relationship between causation and laws, the philosophy of applied m atheatics, and Kant' s philosophy of science.

Theories of Matter: Infinities and Renormalization Leo P. Kadanoff The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter examines the theory underlying the sc ienc e of materials, evaluates the progress in the understanding of the thermodynamic phases of matter, and disc usses c ondensed matter physic s and the idea that c hanges in phase involve the presenc e of a mathematic al singularity. It also argues that the understanding of the behavior of systems at and near phase transitions requires a synthesis between standard statistic al mec hanic al uses of probabilities and c onc epts from dynamic al systems theory. K ey words: sci en ce of materi al s, ph ases of matter, con den sed matter ph y si cs, math emati cal si n gu l ari ty , ph ase tran si ti on s, probabi l i ti es, dy n ami cal sy stems th eory 1. Intro ductio n 1.1 The Discovery and Invention of Materials From the “stone age” through the present day humankind has made use of the materials available to us in the earth and equally to materials we c ould manufac ture for further use. Jared Diamond (1997), for example, in his book Guns, Germs and Steel, has pointed out how c ruc ial metalworking was to the spread of European power. However, it is only in relatively rec ent times that we were able to bring sc ientific understanding of the inner workings of materials to human benefit. The latter half of the nineteenth c entury brought the beginning of two major theoretic al advanc es to the sc ienc e of materials, advanc es that would deepen and grow into the twentieth c entury so that today we c an boast of a fundamental understanding of the main properties of many materials. These advanc es are a theory of statistic al physic s developed initially by Rudolf Clausius (Brush 1976), J. C. Maxwell (Garber, Bush, and Everett 1986), and Ludwig Boltz mann (Brush 1976, 1983) and an understanding of the different phases of matter based in part upon an understanding of the c hanges from one phase1 to another. These c hanges are called phase transitions and our understanding of them is based upon the work of Thomas Andrews (1869), Johannes van der Waals (Levelt-Sengers 1976), and Maxwell. It is the purpose of this essay to develop a desc ription of the development of these basic ideas from the 1870s to the last quarter of the twentieth c entury. Muc h of our first princ iples understanding of materials is based upon the fac t that they are c omposed of many, many atoms, elec trons, and molec ules. This means that we c annot hope or wish to follow any motion of their individual c onstituents, but that instead we must desc ribe their average or typic al properties through some sort of statistical treatment. Further, though we should believe that the properties of these materials are built upon the properties of the c onstituents, we should also rec ogniz e that the properties of a huge number of c onstituents, all working together, might be quite different from the behaviors we might infer from thinking about only a few of these partic les at onc e. P. W. Anderson has emphasiz ed the differenc e by using the phrase “More Is Different” (1972; Ong and Bhatt 2001). Page 1 of 33

Theories of Matter: Infinities and Renormalization There are two surprising differenc es that have dominated the study of materials: irreversibility and the existenc e of sharply distinc t thermodynamic phases. First, irreversibility: even though the basic laws of both c lassic al mec hanic s and quantum mec hanic s are unc hanged under a time reversal transformation, any appropriately statistic al treatment of systems c ontaining many degrees of freedom will not show suc h an invarianc e. Instead, suc h systems tend to flow irreversibly toward an apparently unc hanging state c alled statistical equilibrium. This flow was rec ogniz ed by Clausius in his definition of entropy, detailed by Boltz mann in his gas dynamic definition of the Boltz mann equation, and used by Gibbs (Rukeyser 1942) in his definitions of thermodynamic s and statistic al mec hanic s. It is a behavior that is best rec ogniz ed as a property of a limiting c ase, either of having the number of degrees of freedom become infinite or having an infinite observation time. Time reversal asymmetry is not displayed by any finite system desc ribed over any finite period of time.2 Figure 4.1 Splash and snowflake. This picture is intended to illustrate the qualitative differences between the fluid and solid phases of water. On the left is liquid water, splashing up against its vapor phase. Its fluidity is evident. On the right is a c rystal of ic e in the form of a snowflake. Note the delic ate but rigid struc ture, with its symmetry under the partic ular rotations that are multiples of sixty degrees. This essay is c onc erned with thermodynamic equilibrium resulting from this irreversible flow. It foc uses upon another property of matter that, as we shall see, only fully emerges when we c onsider the limiting behavior as an equilibrium material becomes infinitely large. This property is the propensity of matter to arrange itself into different kinds of struc tures that are amaz ingly diverse and beautiful. These struc tures are c alled thermodynamic phases. Figure 4.1 illustrates three of the many thermodynamic phases formed by water. Water has many different solid phases. Other fluids form liquid c rystals, in whic h we c an see mac rosc opic manifestations of the shapes of the molec ules forming the c rystals. The alignment of atomic spins or elec tronic orbits c an produc e diverse magnetic materials, inc luding ferromagnets, with their substantial magnetic fields, and also many other more subtle forms of magnetic ordering. Our ec onomic infrastruc ture is, in large measure, based upon the various phase-dependent capabilities of materials to carry electrical currents: from the refusal of insulators, to the flexibility of semiconduc- tors, to the substantial carrying capacity of conductors, to the weird resistance-free behavior of superconductors. This flow, and other strange properties of superconductors, are manifestations of the subtle behavior of quantum systems, usually only seen in microscopic properties, but here manifested by these materials on the everyday scales of centimeters and inches. I could go on and on. The point is that humankind has, in part, understood these different manifestations of matter, manifestations that go under the name “thermodynamic phases.” Scientific work has produced at least a partial understanding of how the different phases change into one another. This chapter is a brief desc ription of the ideas c ontained in the sc ienc e of suc h things. As is the c ase with irreversibility, the differenc es among solid, liquid, and gas; the distinc tions among magnetic materials and between them and nonmagnetic materials; and the differences between normal materials and super- fluids are all best understood as distinctions that apply in the limit in which the number of molecules is infinite. For any finite body, these distinctions are blurry with the different cases merging into one another. Only in the infinite limit can the sharp distinction be maintained. Of course, our usual samples of everyday materials contain a huge number of molecules, so the blur in the distinction between different phases is most often too fine for us to discern. However, if one is to set up a theory of these materials, it is helpful to respect the difference between finite and infinite. 1.2 Different Phases; Different Properties Three of the phases of water are illustrated in figure 4.1, whic h depic ts a snowflake (a c rystal of a solid phase of water) and a splashing of the liquid phase. A third phase of water, the low-density phase familiar as water vapor, or steam, exists in the empty-looking region above and around the splashing liquid. Page 2 of 33

Theories of Matter: Infinities and Renormalization Different phases of matter have qualitatively different properties. As you see, ic e forms beautiful c rystal struc tures. So do other solids, eac h with its own c harac teristic shape and form. Eac h c rystal pic ks out partic ular spatial direc tions for its c rystal axes. That selec tion oc c urs bec ause of forc es produc ed by the interac tions of the mic rosc opic c onstituents of the c rystals. Crystalline materials are formed at relatively low temperatures. At suc h temperatures, mic rosc opic forc es tend to line up neighboring molec ules and thereby produc e strong c orrelations between the orientations of c lose neighbors. Suc h c orrelations extend through the entire material, with eac h molec ule being lined up by several neighbors surrounding it, thereby produc ing an ordering in the orientation of the molec ules that c an extend over a distanc e a billion times larger than the distanc e between neighboring atoms. This orientational order thus bec omes visible in the mac rosc opic struc ture of the c rystal, as in figure 4.1, forming a mac rosc opic manifestation of the effec ts of mic rosc opic interac tions. The same materials behave differently at higher temperature. Many melt and form a liquid. The long-range orientational order disappears. The material gains the ability to flow. It loses its spec ial direc tions and gains the full rotational symmetry of ordinary spac e. Many of the mac rosc opic manifestations of matter c an be c harac teriz ed as having broken symmetries. Phases other than simple vapors or liquids break one or more of the c harac teristic symmetry properties obeyed by the mic rosc opic interac tions of the c onstituents forming these phases. Thus, a snowflake's outline c hanges as it is rotated despite the fac t that the molec ules forming it c an, when isolated, freely rotate. 1.2.1 Broken Symmetries and Order Parameters A previous public ation (Kadanoff 2009) desc ribes the development of the theory of phase transitions up to and inc luding the year 1937. In that year, Lev Landau (Landau 1937) put together a theoretic al framework that generaliz ed previously existing mean field theories of phase transitions. In Landau's approac h, a phase transition manifests itself in the breaking of a mathematic al symmetry. This breaking is, in turn, reflec ted in the behavior of an order parameter desc ribing both the magnitude and nature of the broken symmetry. Two different phases plac ed in contact are seen to be distinguished by having different values of the order parameter. For example, in a ferromagnet the order parameter is the vec tor magnetiz ation of the material. Sinc e the spins generate magnetic fields, this alignment is seen as a large time-independent magnetic field vec tor, pointing in some partic ular direc tion. The order parameter in this example is then the vec tor desc ribing the orientation and siz e of the material's magnetiz ation and its resulting magnetic field. Note that the orientation of the order parameter desc ribes the way in whic h the symmetry is broken, while the magnitude of this parameter desc ribes how large the symmetry breaking might be. Other order parameters desc ribe other situations. A familiar order parameter c harac teriz es the differenc e between the liquid and the vapor phases of water. This parameter is simply the mass density, the mass per unit volume, minus the value of that density at the c ritic al point. This parameter takes on positive values in the liquid phase and negative ones in the vapor phase. These phases are c learly exhibited in proc esses of c ondensation, or boiling, or when the two phases stand in c ontac t with one another. In superfluid examples, ones in whic h there is partic le flow without dissipation, a finite frac tion of the partic les are desc ribed by a single, c omplex wave func tion. The order parameter is that wave func tion. The alignment of atomic spins or elec tronic orbits c an produc e diverse magnetic materials, inc luding ferromagnets, with their substantial magnetic fields, and also many subtler forms of magnetic ordering. A c rystal in whic h the magnetiz ation c an point in any direc tion in the three-dimensional spac e c onventionally labeled by the X, Y, and Z axes is termed an “XYZ” ferromagnet. Another kind of ferromagnet is c alled an “XY” system, and is one in whic h internal forc es within the ferromagnet permit the magnetiz ation to point in any direc tion in a plane. The next logic al possibility is one in whic h there a few possible axes of magnetiz ation, and the magnetiz ation c an point either parallel or antiparallel to one of these axes. A simple model desc ribing this situation is termed an Ising model, named after the physic ist Ernst Ising (1925), who studied it in c onjunc tion with his adviser Wilhelm Lenz (Brush 1967; Niss 2005) (see sec tion 2). 1.2.2 Dynamics and Equilibrium We have already mentioned the propensity of c ondensed matter systems to approac h an unc hanging state c alled thermodynamic equilibrium. This approac h c an be either extremely fast or extremely slow, depending upon the Page 3 of 33

Theories of Matter: Infinities and Renormalization situation. When an elec tric al c urrent in a metal is set in motion by an applied voltage it c an take as little as a millionth of a billionth of a sec ond for the c urrent to reac h c lose to its full value. On the other hand, impurities may take years to diffuse through the entire volume of a metal. Bec ause of this very broad range of timesc ales, and bec ause of the wide variety of mec hanisms for time-dependenc e, dynamic s is a very c omplex subjec t. Equilibrium is muc h simpler. The equilibrium state of a simple material is c harac teriz ed by a very few parameters desc ribing the thermodynamic environment around the material. To define the thermodynamic state of a c ontainer filled with water, one needs to know its temperature, the volume of the c ontainer, and the applied pressure. One c an then direc tly c alc ulate the mass of the water in the volume using the data from what is c alled the equation of state. By the time our story begins, in the early twentieth c entury, kinetic theory will be well-developed as a desc ription of c ommon gases and liquids. That theory inc ludes the statement that molec ules in these fluids are in rapid motion, with a kinetic energy proportional to the temperature. Thermodynamics provides a more broadly applic able theory, brought close to its present state by J. Willard Gibbs in 1878 (Gibbs 1961; Rukeyser 1942 pp. 55–371). That disc ipline provides relations among the properties of many-partic le systems based upon c onservation of energy and the requirement that the system always develops in the direc tion of thermodynamic equilibrium. Thermodynamic c alc ulations are based upon thermodynamic func tions, for example, the Helmholtz free energy used in this essay. This func tion depends upon the material's temperature, volume, and the number of partic les of various types within it. One c an c alc ulate all kinds of other properties by c alc ulating derivatives of the free energy with respec t to its variables. The free energy has the further important property that the ac tual equilibrium c onfiguration of the system will produc e the minimum possible value of the free energy. Any other c onfiguration at that temperature, in that volume, with those c onstituents will nec essarily have a higher free energy. To understand phase transitions, one has to go beyond thermodynamic s, whic h c onc erns itself with relations among mac rosc opic properties of materials, to the subjec t of statistical mechanics that defines the probabilities for observing various phenomena in a material in thermodynamic equilibrium, starting from a mic rosc opic desc ription of the behavior of the materials’ c onstituents. Statistic al mec hanic s further differs from thermodynamic s in that the latter treats only summed or gross properties of matter, while the former looks to the individual c onstituents and asks about the relative probabilities for their different c onfigurations and motions. To use statistic al mec hanic s, in princ iple, all one needs to know is a func tion c alled the Hamiltonian, whic h gives the system's energy as a func tion of the c oordinates and momenta of the partic les.3 In prac tic e, for many-partic le systems, the ac tual c alc ulations are suffic iently hard so that in large measure they only bec ame possible after World War II. Statistic al mec hanic s is c lassic ally defined by using a phase space given by the momenta and c oordinates of the partic les in the system. The state of a partic ular c lassic al system is then defined by giving a point in that spac e. The basic s of statistic al mec hanic s were put forward by Boltz mann (Cerc ignani 1998 p. 8, c h. 7; Uffink 2005; Gallavotti 2008) and then c learly stated by Gibbs (1902) in essentially the same form as it is used today. It desc ribes the probability for finding a c lassic al system in a partic ular c onfiguration, c, in the phase spac e. The probability of finding the system in a small volume of phase spac e, dΩ, around c onfiguration c is given by (1) Here, H (c) is the energy in this configuration, T is the absolute temperature in energy units, and F is a constant, whic h turns out to be the free energy. Sinc e the total probability of all c onfigurations of the system must add up to unity, one can determine the free energy by (2) where the integral c overs all of phase spac e. This probability formulation, the basis of all of statistic al mec hanic s, is known as the Maxwell-Boltz mann distribution, to most physic ists and c hemists, or the Gibbs measure, to most mathematic al sc ientists. This probability of Eq. (1) desc ribes a c ollec tion of identic al materials arrayed in different c onfigurations. Gibbs c alls the “ensemble” of c onfigurations thus assembled a “c anonic al ensemble4 ”. Page 4 of 33

Theories of Matter: Infinities and Renormalization 1.2.3 Phase Transitions A treatment of phase transitions may properly start with the 1869 experimental studies of Andrews (1869), who investigated the phase diagram of c arbon dioxide and thereby disc overed the qualitative properties of the liquid– gas phase transition. His results, as illuminated by the theoretic al work of van der Waals and Maxwell (Levelt- Sengers 1976), are shown in figure 4.2. This plot gives the pressure as it depends upon volume in a c ontainer with a fixed number of partic les. Eac h c urve shows the behavior of the pressure at a given value of the temperature. The two c urves at the bottom show results quite familiar from our experienc e with water. At high pressures one has a liquid and the liquid is squeez ed into a relatively small volume. Sinc e the density of the liquid is the (fixed) number of partic les divided by a varying volume, this high-pressure region is one of high density. The c urve moves downward showing a reduc ed pressure as the liquid is allowed to expand. At a suffic iently low pressure, the liquid starts to boil and thereby further reduc e its density until a suffic iently low volume is reac hed so that it has attained the density of vapor. The boiling is what is c alled a first-order phase transition. The boiling oc c urs at a c onstant pressure and has liquid and vapor in c ontac t with one another. Then after the vapor density is reac hed, additional expansion produc es a further reduc ed density. At somewhat higher temperatures this same sc enario is followed, on a higher c urve, exc ept that the region of boiling and its c onnec ted jump in density is smaller. Andrews's big disc overy was that at a suffic iently high temperature, the jump in density disappears and the fluid goes from high pressure to low without a phase transition. This disappearanc e of the first-order phase transition oc c urs at what is c alled a critical point. The disappearanc e itself is c alled a continuous phase transition. 1.2.4 The First Mean Field Theory In his thesis of 1873, Johannes van der Waals put together an approximate theory of the behavior of liquids using arguments based in ideas of the existenc e of molec ules. The very existenc e of molec ules was an idea then current, but certainly not proven. Van der Waals started from the known relation between the pressure and the volume of a perfec t gas, that is, one that has no interac tions between the molec ules. Expressed in modern form, the relation is (3) Figure 4.2 Cartoon PVT diagram for water. Each curve describes how the pressure depends upon volume for a fixed temperature. Note the figures for c ritic al temperature and pressure on this diagram. They apply to water. The c orresponding figures for c arbon dioxide are 31.1° C and 73 atmospheres = 7.2 megapasc als. These values are more easily ac c essible to experiments than the ones for water. Here, p is the pressure, V is the volume of the c ontainer, N is the number of molec ules within it, and T is the temperature expressed in energy units.5 This equation of state relates the pressure, temperature, and density of a gas in the dilute-gas region in whic h we may presume that interac tions among the atoms are quite unimportant. It says that the pressure is proportional to the temperature, T, and to the density of partic les, N/V. This result is inferred by asc ribing an average kinetic energy to eac h molec ule proportional to T and then c alc ulating the transfer of momentum per unit area to the walls. The pressure is this transfer per unit time. Of c ourse, Eq. (3) does not allow for any phase transitions. Page 5 of 33

Theories of Matter: Infinities and Renormalization Two c orrec tions to this law were introduc ed by van der Waals to estimate how the interac tions among the molec ules would affec t the properties of the fluid. First, he argued that the molec ules c ould not approac h eac h other too c losely bec ause of an inferred short-ranged repulsive interac tion among the molec ules. This effec t should reduc e the volume available to the molec ules by an amount proportional to the number of molec ules in the system. For this reason, he replac ed V in Eq. (3) by the available volume, V – Nb, where b would be the exc luded volume around eac h molec ule of the gas. The sec ond effec t is more subtle. The pressure, p, is a forc e per unit area produc ed by the molec ules hitting the walls of the c ontainer. However, van der Waals inferred that there was an attrac tive interac tion pulling eac h molec ule toward its neighbors. This attrac tion is the fundamental reason why a drop of liquid c an hold together and form an almost spheric al shape. As a molec ule moves toward a wall, it is pulled bac k and slowed by the molec ules left behind. Bec ause of this reduc ed speed, it imparts less momentum to the walls than it would otherwise. The equation of state c ontains the pressure as measured at the wall, p. This pressure is the one produc ed inside the liquid, NT/(V – Nb), minus the c orrec tion term c oming from the interac tion between the molec ules near the walls. That c orrec tion term is proportional to the density of molec ules squared. In symbols Van der Waals’ c orrec ted expression for the pressure is thus (4) Here, a and b are parameters that are different for different fluids and N/V is the density of molec ules. Eq. (4) is the widely used van der Waals equation of state for a fluid. Bec ause it takes into ac c ount average forc es among partic les, we desc ribe it and similar equations as the result of a mean field theory. This equation of state c an be used to c alc ulate the partic le density, ρ = N/V, as a func tion of temperature and pressure. It is a c ubic equation for ρ and thus has at most three solutions. 1.2.5 Maxwell's Improvement Figure 4.3 PVT curves predicted by the theory set up by van der Waals. The fluid is mechanically unstable whenever the pressure inc reases as the volume inc reases. The equation of state proposed by van der Waals is plotted in figure 4.3. Eac h c urved line shows the dependenc e of pressure on volume. This equation of state has a major defec t: it shows no boiling region. Worse yet, it c ontains regions in whic h, at fixed temperature and numbers of partic les, the pressure inc reases as the volume inc reases. This situation is unstable. If the fluid finds itself in a region with this kind of behavior, the forc es within it will c ause it to separate into two regions, one at a high density the other at a lower one. In fac t, exac tly this kind of separation does happen in the boiling proc ess in whic h a lower-density vapor is in c ontac t with a high-density liquid. The instability just desc ribed is termed a mec hanic al instability. It c an be triggered by a fluc tuation in whic h a piec e of fluid acquires a density slightly different from that of the surrounding fluid elements. J. C. Maxwell (1874; 1875) in 1874 and 1875 rec ogniz ed this instability and also the somewhat bigger region of thermodynamic instability against larger fluctuations. Maxwell identified this bigger region of instability with boiling and drew a phase diagram like that in figure 4.2. Note that this figure has a c ompletely flat portion of the c onstant temperature lines to represent the Page 6 of 33

Theories of Matter: Infinities and Renormalization predic ted boiling of Maxwell's theory. We shall hear more of this Maxwell c onstruc tion in sec tion 4. Maxwell's result gives a qualitative pic ture of the jump in density between the two phases over a quite wide range of temperatures. For the purposes of this essay, however, the most important region is the one near the c ritic al point in whic h the jump is small. Ac c ording to the theory, as the jump in density, ρ = N/V, goes to z ero, it shows a behavior (5) where Tc is the c ritic al temperature and β has the value one half. Andrews's data does fit a form like this, however with a value of the exponent, β, muc h c loser to one third than one half. Later on, this disc repanc y will bec ome quite important. Despite the known disc repanc y between mean field theory and experiment in the region of the c ritic al point, few sc ientists foc used upon this issue in the years in whic h mean field theory was first being developed. There was no theory or model that yielded Eq. (5) with any power different from one half, so there was no foc us for anyone's disc ontent. Thomas Kuhn (1962) has argued that an old point of view will c ontinue on despite evidenc e to the c ontrary if there is no replac ement theory. Following soon after van der Waals, many other sc ientists developed mean field theories, applying them to many different kinds of phase transitions. All these theories have an essential similarity. They foc us upon some property of the many-partic le system that breaks some sort of global symmetry6 . In mean field c alc ulations, the ordering in one part of the system induc es ordering in neighboring regions until, after some time, ordering is spread through the entire system. Thus mean field theory c alc ulations are always desc riptive of symmetry breaking and the induc ed c orrelations that c arry the symmetry breaking through the material. These c alc ulations are then most relevant and immediately useful for the desc ription of the jumps that oc c ur in first-order phase transitions. 1.3 Fluctuations In equilibrium, the material has a behavior that, in a gross examination, looks time-independent. Henc e, many of the phenomena involved may be desc ribed by using time-averages of various quantities. This averaging is the basis of mean-field-theory tec hniques. However, a more detailed look shows fluctuations, that is, time dependenc e, in everything. These fluc tuations will c all for additional c alc ulational tec hniques beyond mean field theory, whic h will be realiz ed with the renormaliz ation group methods desc ribed in sec tion 6. Here, I desc ribe two important examples of fluctuations that arise near phase transitions. 1.3.1 Fluctuations I: Boiling In the proc ess of ordering, typic ally a material will display large amounts of disorder. For example, as the pressure is reduc ed at the liquid– gas c oexistenc e line, a liquid turns into a vapor by an often-violent proc ess of boiling. The boiling produc es bubbles of low-density vapor in the midst of the higher-density liquid. Thus the fluid, whic h is quite homogeneous away from its phase transition, shows a rapidly fluc tuating density in its boiling region. As every c ook knows, one c an reduc e the violenc e of the fluc tuations by making the boiling less rapid. Nonetheless, it remains true that the fluid shows an instability in the direc tion of fluc tuations in its region of boiling in the phase diagram. 1.3.2 Fluctuations II: Critical Opalescence A proc ess, not entirely dissimilar to boiling, oc c urs in the equilibrium fluid near its c ritic al point. Observers have long notic ed that, as we move c lose to the liquid– gas c ritic al point, the fluid, hitherto c lear and transparent, turns milky. This phenomenon, c alled critical opalescence, was studied by Marian Smoluc howski (1908) and Albert Einstein (1910; Pais 1983, p. 100). Both rec ogniz ed that c ritic al opalesc enc e was c aused by the sc attering of light from fluc tuations in the fluid's density. They pointed out that the total amount of light sc attering was proportional to the c ompressibility, the derivative of the density with respec t to pressure.7 They also noted that the large amount of sc attering near the c ritic al point was indic ative of anomalously large fluc tuations in that region of parameters. In this way, they provided a substantial explanation of c ritic al opalesc enc e.8 1.4 Ornstein and Zernike Page 7 of 33

Theories of Matter: Infinities and Renormalization Leonard Ornstein and Frederik Zernike (Ornstein and Zernike 1914) subsequently derived a more detailed theory of c ritic al opalesc enc e. In modern terms, one would say that the sc attering is produc ed by small regions, droplets, of materials of the two different phases in the near-c ritic al fluid. The regions would bec ome bigger as the c ritic al point was approac hed, with the c orrelations extending over a spatial distanc e c alled the c orrelation or c oherenc e length, ξ. Ornstein and Zernike saw this length diverge on the line of coexistence between the two phases of liquid– gas phase transition as the c ritic al point was approac hed in the form (6) with T — Tc being the temperature deviation from c ritic ality Thus, the c orrelation length does go to infinity as the c ritic al point is approac hed. As we shall see below in sec tion 2.2, this result is c ruc ially important to the overall understanding of the critical point. 1.5 Outline of Essay The next sec tion defines the Ising model, a simple and basic model for phase transitions. It then uses that model to desc ribe the extended singularity theorem, whic h desc ribes the relationship between phase transitions, mathematic al singularities in thermodynamic func tions, and c orrelated fluc tuations. Sec tion 3 then defines mean field theories as one way of approaching the theory of phase transitions. The next section describes the 1937 Landau theory as the pinnacle of mean field theory descriptions. But Landau's work also starts the replacement of mean field theory by fluc tuation-dominated approac hes. In that same year, a c onferenc e in Amsterdam exhibited the c onfusion c aused by the c onflic t between mean field theory and the extended singularity theorem. The long series of studies that indic ated a need for supplementing mean field theory is desc ribed in sec tion 5. Sec tion 6 desc ribes the development of a new phenomenology to understand fluc tuations in phase transitions. Kenneth Wilson transformed that phenomenology into a theory by adding ideas described in section 7. The concepts that grew out of that revolution are disc ussed in the final sec tion. 2. The Ising Mo del 2.1 Definition Figure 4.4 Lattice for two-dimensional Ising model. The spins are in the circles. The couplings, K, are the lines. A partic ular site is labeled with an “r.” Its nearest neighbors are shown with an “s.” The Ising model is a c onc eptually simple representation of a system that c an potentially show ferromagnetic behavior. Its name c omes from the physic ist Ernst Ising (1925), who studied it9 in c onjunc tion with his adviser Wilhelm Lenz (Brush 1967). Real ferromagnets involve atomic spins plac ed upon a lattic e. The eluc idation of their properties requires a diffic ult study via the band theory of solids. The Ising model is a shortc ut that c atc hes the main qualitative features of the ferromagnet. It puts a spin variable upon eac h site, labeled by r, of a simple lattic e. (See figure 4.4.) Eac h spin variable, σr, takes on values plus or minus one to represent the possible direc tions that might be taken by a particular component of a real spin upon a real atom. The sum over c onfigurations is a sum over all these possible values. The Hamiltonian for the system is the simplest representation of the fact that neighboring spins interact with a dimensionless coupling strength, K, and a Page 8 of 33

Theories of Matter: Infinities and Renormalization dimensionless c oupling to an external magnetic field, h. The Hamiltonian is given by (7) where the first sum is over all pairs of nearest neighboring sites, and the sec ond is over all sites. The ac tual c oupling between neighboring spins, with dimensions of an energy, is often c alled J. Then K = −J/T. In turn, h is proportional to the magnetic moment of the given spin times the applied magnetic field, all divided by the temperature. Sinc e lower values of the energy have a larger statistic al likelihood, the two terms is Eq. (7) reflec t respec tively a tendenc y of spins to line up with eac h other and also a tendenc y for them to line up with an external magnetic field. 2.2 The Extended Singularity Theorem Partic le spin is c ertainly a quantum mec hanic al c onc ept. There is no simple c orrespondenc e between this c onc ept and anything in c lassic al mec hanic s. Nonetheless, the c onc ept of spin fits smoothly and easily into the Boltz mann- Gibbs formulation of statistic al mec hanic s. When spins are present, the statistic al sum in Eq. (2) inc ludes a quantum summation over eac h spin-direc tion. In the Ising c ase, when the only variable is σ, standing for the z- component of the spin, that summation operation is simply a sum over the two possible values plus one and minus one of eac h spin at eac h lattic e site. For the Ising model, the integral over c onfigurations in Eq. (2) is replac ed by a sum over the possible spin values, thus making the result have partic ularly simple mathematic al properties. Take the logarithm of that equation and find (8) On the right-hand side of this equation one finds a simple sum of exponentials. This is a sum of positive terms, and it gives a result that is a smooth func tion of the parameters in eac h exponential, spec ific ally the dimensionless spin-c oupling, K, and the dimensionless magnetic field h. A logarithm of a smooth func tion is itself a smooth func tion. Therefore, it follows direc tly that the free energy, F, is a smooth func tion of h and K. The reader will notic e that this smoothness seems to c ontradic t our definition of a phase transition, the statement that a phase transition is a singularity, that is, failure of smoothness, in some thermodynamic quantity. This seeming c ontradic tion is the key to understanding phase transitions. No sum of a finite number of smooth terms c an be singular. However, for large systems, the number of terms is the sum grows quite rapidly with the siz e of the system. When the system is infinite, the number of terms is infinite. Then singularities c an arise. Thus, all singularities, and henc e all phase transitions, are c onsequenc es of the influenc e of some kind of infinity. Among the likely possibilities are infinite numbers of partic les, infinite volumes, or —more rarely— infinitely strong interac tions. Real c ondensed matter systems often have large numbers of partic les. A c ubic c entimeter of air c ontains perhaps 102 0 partic les. When the numbers are this large, the systems most often behave almost as if they had an infinity of partic les. Page 9 of 33

Theories of Matter: Infinities and Renormalization Figure 4.5 Cartoon view of a singularity in a phase transition. The magnetic susceptibility, the derivative of the magnetization with respec t to the magnetic field, is plotted against temperature for different values of N. The thic k solid c urve is shows the susc eptibility in an infinite system. The dashed c urves apply to systems with finite numbers of partic les, with the higher line being the larger number of partic les. The c ompressibility of the liquid– gas phase transition also shows this behavior. I am going to give a name to the idea that phase transitions only oc c ur when the c ondensed matter system exhibits the effect of some singularity extended over the entire spatial extent of the system. Usually the infinity arises bec ause some effec t is propagated over the entire c ondensed system, that is, over a potentially unbounded distanc e. I am going to c all this result the “extended singularity theorem,” despite the fac t that the argument is rather too vague to be a real theorem. It is instead a slightly imprec ise mathematic al property of real phase transitions.10 This theorem is only partially informative. It tells us to look for a sourc e of the singularity, but not exac tly what we should seek. In the important and usual c ase in whic h the phase transition is produc ed by the infinite siz e of the system, the theorem tells us that any theory of the phase transition should look to things that happen in the far reaches of the system. What things? How big are they? How should one look for them? Will they dominate the behavior near the phase transition or be tiny? The theorem is uninformative on all these points. Sometimes it is very hard to see the result of the theorem. In an Ising or liquid–gas phase transition there is a singularity in the regions just touc hing the c oexisting phases (Adreev 1964; Fisher 1978). This singularity is very weak. One must use indirec t methods to observe or analyz e it. Conversely, near c ritic al points, singularities are very easy to observe and measure. For example, in a ferromagnet, the derivative of the magnetiz ation with respec t to the applied magnetic field is infinite at the c ritic al point. (See figure 4.5.) Figure 4.6 Phase diagram for ferromagnet and Ising model. The jump in magnetization occurs at zero magnetic field. In this representation, the jump region has been reduc ed to a line running from zero temperature up to the c ritic al point. By looking at simulations of finite-siz ed Ising systems one c an see how the infinite siz e of the system enters the susc eptibility. Figure 4.5 is a set of plots of susc eptibility versus temperature in an Ising system with a vanishingly small positive magnetic field. The different plots show what happens as the number of partic les inc reases toward infinity. As you c an see, the finite N c urves are smooth, but the infinite-N c urve goes to infinity. This infinity is the singularity. It does not exist for any finite value of N. However, as N gets larger, the finite-N result approac hes the infinite-N curve. When we look at a natural system, we tend to see phase transitions that look very sharp indeed, but are ac tually slightly rounded. However, a c onc eptual understanding of phase transitions requires that we c onsider the limiting, infinite-N, c ase. Now we c an see the importanc e of the Ornstein-Zernike infinity in the c orrelation length. This last infinity ac c ompanies and c auses the infinity in the susc eptibility, and both of these then require an infinite system for their realiz ation. Figure 4.6 is the phase diagram of the Ising model. The x axis is the magnetic field; the y axis is the temperature. This phase diagram applies when the lattic e is infinite in two or more dimensions. There is no phase transition for lower dimensionality. 3. Mo re Is the Same Page 10 of 33

Theories of Matter: Infinities and Renormalization This sec tion desc ribes mean field theory, whic h forms the basis of muc h of modern many-partic le physic s and field theory. So far, we said that an infinite statistic al system sometimes has a phase transition, involving a disc ontinuous jump in a quantity c alled the order parameter. But we have given no indic ation of how big the jump might be, nor of how the system might produc e it. Mean field theory provides a partial and approximate answer to that question. We begin with the statistic al mec hanic s of one spin in a magnetic field. Then, we extend this one-spin disc ussion to desc ribe how many spins work together to produc e ferromagnetism. 3.1 One Spin A single spin in a magnetic field c an be desc ribed by a simplific ation of the Ising Hamiltonian of Eq. (7), −H/T = hσ. As before, σ is a c omponent of the spin in the direc tion of the magnetic field. This quantum variable takes on two values ±1, so that probability distribution of Eq. (1) gives the average value of the spin as (9) (In general, we write the statistic al average of any quantity, q, as 〈 q 〉.) 3.2 Many Spins; Mean Fields The very simple result, Eq. (9), appears again when one follows Pierre Curie (1895) and Pierre Weiss (1907) in their development of a mean field theory of ferromagnetism. Translated to the Ising c ase, their theory would ask us to c onc entrate our attention upon one Ising variable, say the one at r. We would then notic e that this one spin would see a field with the value (10) where h(r) is the dimensionless magnetic field at r and the sum c overs all the spins with positions, s, sitting at nearest neighbor sites to r. To get the mean field result, replac e the ac tual values of all the other spins, but not the one at r, by their average values and find, by the same c alc ulation that gave Eq. (9), a result in whic h the average is once more (11a) but now the ac tual field is replac ed by an effec tive field (11b) 3.3 Mean Field Results Given this c alc ulation of basic equations for the loc al magnetiz ation, 〈 σr 〉, we c an go on to find many different aspec ts of the behavior of this mean field magnet. We notic e that when h is independent of position, the equation for 〈 σ 〉 has a c ritic al point, that is, an ambiguity in its solution, at z ero magnetic field and Kz = 1. When we expand the equations around that c ritic al point, we c an find that the magnetiz ation obeys a c ubic equation like that of the van der Waals theory (or equally the Landau theory as desc ribed below in sec tion 4.1. Thus, there is a full and c omplete c orrespondenc e between the van der Waals theory of the liquid– gas transition and the ferromagnetic mean field theory near its c ritic al point, as one c an see by c omparing figure 4.2 with figure 4.6. A brief c alc ulation shows that in mean field theory the magnetic susc eptibility, the derivative of the magnetiz ation with respec t to magnetic field, diverges as 1/| T − Tc | near the c ritic al point. The analogy just mentioned between the liquid−gas system and the magnetic phase transition makes this magnetic susc eptibility the direc t analog of the c ompressibility. Please rec all that the c ompressibility has an infinity that was used by Einstein to explain c ritic al opalesc enc e. (See sec tion 1.3.2). On the other hand, Ornstein and Zernike c alc ulated the fluid analog of the more disaggregated quantity (12) Page 11 of 33

Theories of Matter: Infinities and Renormalization c alled the spin c orrelation func tion. A sum over all lattic e sites, s, of this c orrelation func tion will give the susc eptibility. Then g c an be evaluated from the equations of mean field theory ((Kadanoff 2000), p. 232) as (13) in the simplest c ase: three dimensions, h = 0, t small but greater than z ero, and separation distanc e large c ompared to the lattic e spac ing, a. In Eq. (13), ξ is the c orrelation length that desc ribes the range of influenc e of a c hange in magnetic field. Its value, given by Eq. (6), shows that the c orrelation length diverges as c ritic ality is approac hed. This behavior is an expec ted c onsequenc e of the extended singularity theorem, whic h asks for infinite ranges of influenc e at phase transitions. We previously argued that the extended singularity theorem c alled for fluc tuations extending over large distanc es. Indeed that c all is prec isely answered by Eq. (13) and Eq. (6). A theorem of statistic al mec hanic s relates the c orrelation func tion to spin fluc tuations via (14) The right-hand side of this expression relates g to the deviations of the spins at r and s from their averages values. Ac c ording to Eq. (13), these fluc tuations have c orrelations that persist over distanc es c omparable to the length, ξ, whic h c an then go to infinity as c ritic ality is approac hed. Note the sc aling of the spin c orrelation func tion. For relatively small values of the distanc e, the c orrelation func tion in Eq. (13) has a form in whic h g varies as one divided by the distanc e. It is c onventional to desc ribe this c orrelation func tion, varying as | r − s| −2 x, by saying that there are two loc al quantities c ontributing to the c orrelation and then saying that eac h sc ales as distanc e to the power x. Therefore, in this c ase, the index going with the order parameter is x = 1/2. 3.4 Representing Critical Behavior by Power Laws The reader will, no doubt, have notic ed the appearanc e of “power laws” in the desc ription of behavior near c ritic al points. In these laws, some c ritic al property is written as a power of a quantity that might bec ome very large or very small, as for example, magnetiz ation = c onstant × tβ. So far, we have seen laws like this in the behavior of the order parameter (Eq. (5)), the c orrelation length (Eq. (6)), the magnetic susc eptibility (figure 4.5), and the c orrelation func tion (Eq. (13)). Why does this power func tion appear repeatedly? All of this singular behavior is rooted in the fac t that phase transitions produc e a variation over a tremendous range of length sc ales. For example, the basic interac tions driving most phase transitions oc c ur on a length sc ale desc ribed by the distanc e between atoms or molec ules, that is some frac tion of a nanometer (10−9 meters). On the other hand, we observe and work with materials on a c harac teristic length sc ale of c entimeters (10−2 meters). The c ruc ial issue in phase transitions is how the material interpolates phenomena over this tremendous length sc ale. The answer is roughly speaking that all the physic al quantities mentioned follow the c hanges in the length sc ale. As we shall see in sec tion 7.2, in renormaliz ation c alc ulations, the c hanges of the length sc ale in turn follow from multiplic ative laws. To get to a tremendous c hange in length sc ale, ℓ, one puts many small steps ℓ1,ℓ2 ,ℓ3 , … into the renormaliz ation c alc ulation and the big c hange is produc ed by the multiplic ation of these fac tors, This kind of behavior is explic itly built into renormaliz ation c alc ulations.11 Sc ale transformation is a symmetry operation. It desc ribes an underlying symmetry of nature in whic h every sc ale —kilometer, c entimeter, nanometer—is equally good for desc ribing nature's basic laws. Whenever a physic al phenomenon reflec ts a symmetry operation, observed physic al quantities must transform under symmetry operations as mathematic al representations of that symmetry. That is why we use sc alars, vec tors, and tensors to desc ribe quantities that obey, say, the usual rotational symmetry. The same thing works for sc ale transformations. Here, power laws reflec t the symmetries built into multiplic ation operations. The physic al quantities behave as powers, ℓx, where x c an be rational or irrational, positive or negative, or indeed z ero. In the last c ase, the limiting behavior is that of a logarithm instead of a power, as is ac tually obtained in the heat c apac ity of the Onsager solution (Onsager 1944) of the two-dimensional Ising model. Page 12 of 33

Theories of Matter: Infinities and Renormalization The wide range of length sc ales also applies in partic le physic s where the basic sc ales for interac tions may be vastly different from the sc ale at whic h observations are performed. Thus, in partic le physic s it is also true that renormaliz ation and sc aling have to interpolate behaviors over very large length sc ales. Whenever one has a power law, say 〈 σ 〉 = (−t)β, one has a power, here β. This power is c alled a “c ritic al exponent” or a “c ritic al index.” During the many years in whic h c ritic al behavior has been a subjec t of sc ientific study, many human-years of sc ientific effort have been devoted to the ac c urate determination of these indic es. Sometimes sc ientists c omplained that this effort was misplac ed. After all, there is little insight to be obtained from the statement that β (the index that desc ribes the jump in the liquid−gas phase transition) has the value 0.31 versus 0.35 or 0.125 or 0.5. But these various values c an be obtained from theories that give a direc t c alc ulation of c ritic al quantities or related them one to another. The c alc ulations or relations c ome from ideas with c onsiderable intellec tual c ontent. Finding the index-values then gave an opportunity to c hec k the theory and see whether the underlying ideas were sound. Thus, the small industry of evaluating c ritic al indic es supports the basic effort devoted to understanding c ritic al phenomena. 4. The Year 1937: A Revo lutio n Begins 4.1 Landau's Generalization Lev Landau followed van der Waals, Pierre Curie, and Ehrenfest in notic ing a deep c onnec tion among different phase transition problems (Daugherty 2007). Landau translated this observation into a mathematic al theory in a novel and interesting way. Starting from the rec ognition that, in the neighborhood of a c ritic al point, eac h phase transition was a manifestation of a broken symmetry, he used the order parameter to desc ribe the nature and the extent of symmetry breaking (Landau 1937). Landau generaliz ed the work of others by writing the free energy as an integral over all spac e of an appropriate func tion of the order parameter. The dependenc e upon r indic ates that the order parameter is c onsidered to be a func tion of position within the system. In the simplest c ase, desc ribed above, the phase transition is one in whic h the order parameter, say the magnetiz ation, c hanges sign.12 In that c ase, the appropriate free energy takes the form (15) where A,B,C,… are parameters that desc ribe the partic ular material and Ψ(r) is the order parameter at spatial position r. In rec ognition of the delic ac y of the c ritic al point, eac h term goes to z ero more rapidly than Ψ(r)2 as c ritic ality is approac hed. The next step is to use the well-known rule of thermodynamic s that the free energy is minimiz ed by the ac hieved value of every possible mac rosc opic thermodynamic variable within the system. Landau took the magnetiz ation density at eac h point to be a thermodynamic variable that c ould be used to minimiz e the free energy. Using the calculus of variations one then gets an equation for the order parameter: (16) One would get a result of prec isely this form by applying the mean field theory magnetiz ation equation near the c ritic al point. The B-term is identified by this c omparison as being proportional to the temperature deviation from criticality, B = −At/2. In some sense, of c ourse, Landau's c ritic al point theory is nothing new. All his results are c ontained within the earlier theories of the individual phase transitions. However, in another sense his work was a very big step forward. By using a single formulation that c ould enc ompass all c ritic al phenomena with a given symmetry type, he pointed out the c lose similarity among different phase transition problems. And indeed in the modern c lassific ation of phase transition problems (Kadanoff et al. 1967), the two main elements of the c lassific ation sc heme are the symmetry of the order parameter and the dimension of the spac e. Landau got the first one right but not, at least in this variational formulation, the sec ond c lassifying feature. On the other hand, Landau's inc lusion of the spac e gradients that then brought together the theory's spac e dependenc e and its thermodynamic behavior also seems, from a present-day perspec tive, to be right on. Page 13 of 33

Theories of Matter: Infinities and Renormalization 4.2 Summary of Mean Field Theories As already mentioned, Landau's 1937 result provides a kind of mean field theory that agrees in all essential ways with the results of the main previous workers. The only differenc e is that Landau produc ed a spec ializ ed theory intended to apply mostly to the region near the c ritic al point. From the point of view of the disc ussion that will follow the main points of his theory are: • Universality. The Landau theory gives an equation for the order parameter as a func tion of the thermodynamic parameters (e.g., t and h) that is universal: it only depends upon the kind of symmetry reflec ted in the ordering. • Symmetry. A first-order phase transition is often, but not always, a reflec tion of a c hange in the basic symmetry of the c ondensed system. • Interac tions. This symmetry c hange is usually c aused by loc al interac tions among the basic c onstituents of the system. • Sc aling. The results depend upon simple ratios of the thermodynamic parameters raised to powers. For example, in the ferromagnetic transition all physic al quantities depend upon the ratio t3 /h2 . In subsequent theories, the restric tion to simple powers will disappear. • Order parameter jump. At the first-order phase transition, there is a disc ontinuous jump in the order parameter. As the c ritic al point is approac hed, the jump goes to z ero with c ritic al index β = 1/2 as in Eq. (5). • Correlation length. The c orrelation length goes to infinity at c ritic ality as in Eq. (6) with an index ν = 1/2. As we shall see, for many purposes, the mean field theories have been replac ed by a renormalization group theory of phase transitions. The qualitative properties of mean field theory, like universality and sc aling, have been retained. On the other hand, all the quantitative properties of the theory, for example, the values of the c ritic al indic es, have been replac ed. 4.3 Away from Corresponding States—Toward Universality Landau's c alc ulation represented the high-water mark of the c lass of theories desc ribed as “mean field theories.” He showed that all of them c ould be c overed by the same basic c alc ulational method. They differed in the symmetries of the order parameter, and different symmetries c ould give different outc omes. However, within one kind of symmetry the result was always the same. This uniform outc ome was very pleasing for many students of the subjec t, partic ularly so for the physic ists involved. We physic ists espec ially like mathematic ally based generaliz ations and Landau had developed an elegant generaliz ation, whic h simplified a c omplex subjec t. However, Landau's uniformity was different from the theoretic al idea of uniformity that had c ome before him. Earlier work had been based upon the idea that different fluids have an almost identic al relation expressing the dependenc e of their pressure upon temperature and density. This idea is c alled the “princ iple of c orresponding states.” This princ iple of c orresponding states had broad support among the sc ientists working on phase transitions. Starting with van der Waals, c ontinuing with the work of Einstein (Pais 1983, p. 57), George Uhlenbec k, and E. A. Guggenheim (1945), work on phase transitions was inspired by the aim and hope that the phase diagrams of all fluids would be essentially alike. However, Landau's work marked a new beginning. His method would apply only near a c ritic al point and his version of c orresponding states c ould be expec ted to apply only in this region. So Landau deepened the theory but implic itly also narrowed its domain of applic ation to a relatively small region of the phase diagram. This was the start of a new point of view, whic h we shall see develop in the rest of this essay. The new point of view would c ome with a new voc abulary so that instead of c orresponding states people would begin to use the word “universality” (Kadanoff 1990). 4.4 Statistical Confusion: A Meeting in the Netherlands The extended singularity theorem (see sec tion 2) presents both an opportunity and a c hallenge for understanding phase transitions. The theorem is self-evident in the c ase of the Ising model with its simple sums and exponentials. Page 14 of 33

Theories of Matter: Infinities and Renormalization It is less obviously true for the statistic al mec hanic s of the liquid– gas phase transition sinc e, in this c ase, the c alc ulation of the free energy inc ludes integrals and also unbounded potentials. However, the theorem remains true for that transition. Thus, the theorem would then demand an infinite system for a sharp liquid–gas phase transition. On the other hand, the van der Waals mean field argument would, for example, give a sharp phase transition in small systems. This c ontradic tion might serve as a c onfusing element in the development of a theory of phase transitions. The c ontradic tion was as old as the first definitions of statistic al mec hanic s and phase transitions, but was apparently not disc ussed for many years. It might well, however, have c ome up at a 1937 meeting held in Amsterdam to c elebrate the c entenary of van der Waals's birth. Hendrik Kramers, George Uhlenbec k, and Peter Debye were all present at that oc c asion. Ac c ording to Uhlenbec k (1978), at that meeting Kramers pointed out that the sharp singularity of a phase transition c ould only oc c ur in a system with some infinity built in and, for that, that an infinite system is required. Then the van der Waals theory's predic tion of a phase transition in a finite system c ould be viewed as a grave failure of mean field theory and maybe even of statistic al mec hanic s. E. G. D. Cohen desc ribed material by Uhlenbec k (Cohen n.d.): “Apparently the audienc e at this van der Waals memorial meeting in 1937, c ould not agree on the above question, whether the partition func tion c ould or c ould not explain a sharp phase transition. So the c hairman of the session, Kramers, put it to a vote.” The proposition was “Can statistic al mechanics describe the liquid state?” The meeting is said to have split 50–50, with Debye (!) voting no! Clearly half the people at that meeting were wrong. Seventy plus years later one c an see the right answer, in c lose analogy to our understanding of irreversibility. Infinite siz e is required for a sharp phase transition, but a large system c an very well approximate the behavior of the infinite system. Finite siz e slightly rounds off and modifies the sharp c orners shown in the plot of figure 4.2. Conventional statistic al mec hanic s, following the path begun by Andrews, van der Waals, and Maxwell, c an desc ribe quite well what happens in the liquid region, espec ially if one stays away from the c ritic al region and from boiling. In fac t, there are theories, inc luding one by John Weeks, David Chandler, and Hans Anderson (1917), that do a good job of desc ribing the liquid region of the fluid. However, the extended singularity theorem does have its effec ts. There are indeed singularities near the first-order transition (Andreev 1964; Fisher 1967). These singularities are very weak in the Ising and liquid–gas transition, but will be stronger when an unbroken symmetry remains after the symmetry breaking of the first-order transition. (We see such a residual symmetry in the Heisenberg model of ferromagnetism.) Also, there are quite strong singularities in the neighborhood of the c ritic al point, not c orrec tly desc ribed by mean field theory. All these singularities are c onsequenc es of fluc tuations, whic h are not inc luded in the mean field approac h. However, the Amsterdam meeting was quite right, in my view, to be disquieted by the applic ability of statistic al mec hanic s. But they foc used upon the wrong part of the phase diagram. The liquid region is desc ribed c orrec tly by statistic al mec hanic s. But this theory does not work well in the two-phase, “boiling” region of figure 4.2. Here the fluc tuations entirely dominate and the system sloshes between the two phases. The behavior of the interfac e that separates the phases is determined by delic ate effec ts of dynamic s and previous history and by hydrodynamic effec ts inc luding gravity, surfac e tension, and the behavior of droplets. Henc e, the direc t applic ation of statistic al mec hanic s is fraught with diffic ulty prec isely in the midst of the phase transition. Thus, the extended singularity theorem suggests that a new theory is required to treat all the fluc tuations appearing near singularities. 5. Beyo nd (o r Beside) Mean Field Theo ry Weaknesses of mean field theory began to bec ome apparent to the sc ientific c ommunity immediately after Landau's statement of the theory in its generaliz ed form. This sec tion will desc ribe the proc ess of displac ement of mean field theory, at least for behavior near the c ritic al point, whic h we might say began in 1937 and c ulminated in Kenneth Wilson's enunc iation of a replac ement theory in 1971 (Wilson 1971). Landau's theory provided a standard and a model for theories of general phenomena in c ondensed matter physic s. Looking at Landau's result one might c onc lude that a theory should be as general and elegant as the phenomena it explained. The mean field theories that arose before (and after) Landau's work were partial and inc omplete, in that eac h referred to a partic ular type of system. That was c ertainly nec essary in that the details of the phase diagrams were different for different kinds of systems, but somewhat similar for different materials of the same general kind. Landau's magisterial work swept all these diffic ulties under the rug and for that reason c ould not apply to the whole Page 15 of 33

Theories of Matter: Infinities and Renormalization phase diagram of any given substanc e. Thus, if Landau were to be c orrec t he would most likely be so in the region near c ritic ality. Certainly his theory is based upon an order parameter expansion that only is plausible in the c ritic al region. However, prec isely in this region, as we shall outline below, both theoretic al and experimental fac ts contradicted his theory. 5.1 Experimental Facts The ghosts of Andrews and van der Waals might have whispered to Landau that a theory that predic ts β = 1/2 near c ritic ality c annot be c orrec t. In addition, a muc h larger body of early work on fluids had pointed to this c onc lusion. These early data, developed and published by J. E. Versc haffelt (1900) and summariz ed by Levelt-Sengers (1976), touc hed almost every aspec t of the c ritic al behavior of fluids. Versc haffelt partic ularly stresses the inc ompatibility of the data with mean field theory. Figure 4.7 Cartoon sketch of heat capacity in the neighborhood of critical temperature as predicted by mean field theory. The heat c apac ity is higher below Tc bec ause there is an additional temperature dependenc e in the free energy in this region produc ed by a term proportional to the square of the order parameter. These same experimental fac ts appear onc e more in the 1945 work of E. A. Guggenheim (1945), who c ompared data for a wide variety of fluids. He says, “The princ iple of c orresponding states may safely be regarded as the most useful by-produc t of van der Waals’ equation of state. While this [van der Waals] equation of state is rec ogniz ed to be of little or no value, the princ iple of c orresponding states as c orrec tly applied is extremely useful and remarkably ac c urate.” He examined data for seven fluids on the line of the liquid−gas phase transition and fit the data to a power law with β = 1/3, rather than the mean field value β = 1/2. The latter value c learly does not work; the former fits reasonably well. Thus “c orresponding states” rec eives support in this region, but not mean field theory per se. But neither Guggenheim nor Heike Kamerlingh Onnes (Levelt-Segers 1976) before him was ready to rec eive information suggesting that behavior in the c ritic al region was spec ial, so that the former rejec ted mean field theory while the latter ac c epted it with reservations as to its quantitative ac c urac y. Later, near-c ritic al data on heat c apac ity, the derivative of average energy with respec t to temperature, bec ame available. Mean field theory predic ts a disc ontinuity in the c onstant volume heat c apac ity as in figure 4.7. L. F. Kellers (1960) looked at the normal fluid to superfluid transition in helium-4 (see figure 4.8.) The data on this phase transition seemed to support the view that the heat c apac ity diverges weakly, perhaps as a logarithm of \\T − Tc \\, as c ritic ality is approac hed. Similar heat c apac ity c urves were observed by Alexander Voronel’ (Bagatskii, Voronel’, and Gusak 1963; Voronel’ et al. 1964) in the liquid– gas transition of c lassic al gases and in further work in helium (Moldover and Little 1965). 5.2 Theoretical Facts Page 16 of 33

Theories of Matter: Infinities and Renormalization Figure 4.8 Heat capacity as measured. This picture, the work of Moldover and Little (1965), shows measured heat c apac ities for the normal– superfluid transition of helium-4, labeled as Tλ, and the liquid– vapor transition of helium-3 and helium-4. Note that all three heat c apac ities seem to spike at the c ritic al point, in c ontrast to the predic tion of mean field theory. As we have seen, experimental evidenc e suggested that mean field theory was inc orrec t in the c ritic al region. A further strong argument in this direc tion c ame from Lars Onsager's exac t solution (1944) of the two-dimensional Ising model, followed by C. N. Yang's c alc ulation (1952) of the z ero-field magnetiz ation for that model. Onsager's result for the heat c apac ity diverged as the logarithm of T − Tc as did the experimental observations, as shown, for example, in figure 4.8, but did not resemble the disc ontinuity of mean field theory.13 Yang's results, for whic h β = 1/8, also disagreed with mean field theory, whic h has β = 1/2. The Onsager solution implies a c orrelation length with ν = 1, whic h is not the mean field value ν = 1/2; see Eq. (6). The most systematic theoretic al disc rediting of mean field theory c ame from the series expansion work of the King's College (London) sc hool, under the leadership of Cyril Domb, Martin Sykes, and—after a time—Mic hael Fisher (Domb 1996; Niss 2005). Rec all that the Ising model is a simplified model that c an be used to desc ribe magnetic transitions. It is desc ribed by a strength of the c oupling between neighboring spins proportional to a c oupling c onstant J. The statistic al mec hanic s of the model is defined by the ratio of c oupling to temperature, spec ific ally K = −J/T. One c an get c onsiderable information about the behavior of these models by doing expansions of quantities like the magnetiz ation and the heat c apac ity in power series in K, for high temperatures, and e−K for low temperatures. The group at King's developed and used methods for doing suc h expansions and then analyz ing them to obtain approximate values of c ritic al indic es like β and ν. The resulting index-values in two dimensions agreed very well with values derived from the Onsager solution. In three dimensions, models on different lattic es gave index values roughly agreeing with experiment on liquids and magnetic materials, but differing substantially from predic tions of mean field theory. This work provided a powerful argument indic ating that mean field theory was wrong, at least near the c ritic al point. It also played a very important role in foc using attention upon that region. Another reason for doubting mean field theory, ironic ally enough, c ame from Landau himself. In 1941, Andrei Kolmogorov developed a theory of turbulenc e based upon c onc epts similar to the ones used in mean field theory, in partic ular the idea of a typic al veloc ity sc ale for veloc ity differenc es over a distanc e r (Kolmogorov 1941). These differenc es would, in his theory, have a c harac teristic siz e that would be a power of r. Landau c ritic iz ed Kolmogorov's theory saying that it did not take into ac c ount fluc tuations (Frisc h 1995), whereupon Kolmogorov modified the theory to make it substantially less similar to mean field theory (1962). 5.3 Spatial Structures The spatial struc ture of mean field theory does not agree with the theorem that phase transitions c an only oc c ur in infinite systems. Mean field theory is based on the alignment of order parameter values at neighboring sites, so that partic les will order if neighboring partic les are ordered also. Any c ollec tion of c oupled spins c an have a mean field theory phase transition. Thus, two spins and a bond are quite suffic ient to produc e a phase transition in a mean field argument like that in sec tion 3.2. On the other hand, the extended singularity theorem insists that the oc c urrenc e of a phase transition requires some sort of infinity, most often the existenc e of an infinite number of interacting parts within the system. Page 17 of 33

Theories of Matter: Infinities and Renormalization As we shall see, what is wrong with mean field theory is that in the c ritic al region the effec t of the average behavior of the order parameter c an be c ompletely swamped by fluc tuations in this quantity. In 1959 and 1960, A. P. Levanyuk and Vitaly Ginz burg desc ribed a c riterion that one c ould use to determine whether the behavior near a phase transition was dominated by average values or by fluc tuations (Levanyuk 1959; Ginz burg 1960). For example, when applied to c ritic al behavior of the type seen in the simplest version of the Ising model, this c riterion indic ates that fluc tuations dominate in the c ritic al region whenever the dimension is less than or equal to four. Henc e, mean field theory is wrong(47) for all the usual c ritic al phenomena in systems with dimension smaller than or equal to four.14 Conversely, this c riterion suggests that mean field theory gives the leading behavior above four dimensions. 6. New Fo ci; New Ideas 6.1 Bureau of Standards Conference So far, the field of phase transitions had lived up perfec tly to Thomas Kuhn's (1962) view of the c onservatism of science. Before World War II, the only theory of phase transitions was mean field theory. No theory or model yielded Eq. (5) with any value of β different from one half. There was no foc us for anyone's disc ontent. For this reason, the mean-field-theory point of view c ontinued on, despite evidenc e to the c ontrary, until a set of events oc c urred that would move the field in a new direc tion. One c ruc ial event was the c onferenc e on c ritic al phenomena held at the US National Bureau of Standards in 1965 (NBS 1965). The late Melville Green was the moving spirit behind this meeting. The point of this c onferenc e was that behavior near the c ritic al point formed a separate body of sc ienc e that might be studied on its own merits, independent of the rest of the phase diagram. In the years just before the c onferenc e, enough work (Domb and Miedema 1964; Fisher 1967; Heller 1967) had been done so that the c onferenc e c ould serve as an inauguration of a new field. We have mentioned the experimental studies of Kellers and of the Voronel’ group. At roughly the same time important theoretic al work was done by Alexander Patashinskii and Valery Pokrovsky (Patashinskii and Pokrovsky 1964), Benjamin Widom (1964; 1965a and b) and myself (1966), whic h would form a basis for a new synthesis. The experimental and theoretic al situation just after the meeting was summariz ed in reviews (Fisher 1967; Heller 1967; Kadanoff et al. 1967). This sec tion begins by reporting on those new ideas and then desc ribes their c ulmination in the work of Kenneth G. Wilson (1971). 6.2 Correlation Function Calculations For many years the Landau group had been using field theory to desc ribe the c ritic al point. Two young theoretic ians, Patashinskii and Pokrovsky, foc used their attention upon the c orrelated fluc tuations of order parameters at many different points in spac e. Their result was simple but powerful. Consider the result of c alc ulating the average of the produc t of m loc al order parameter operators at m different positions, rm, in a system at the c ritic al point. (All differenc es between positions of the operators should be large in c omparison to the distanc e between neighboring sites or the range of forc es.) Compare this average with the same c orrelation func tion c alc ulated at the positions ℓ × rm. All that has been done is to c hange the length-sc ale on whic h the c orrelations have been defined. Patashinskii and Pokrovsky then argued that this c hange in sc ale was an invarianc e of the system so that the two c orrelation func tions will have prec isely the same struc ture (Patashinskii and Pokrovsky 1964), and differ by a fac tor ℓ−mx. A similar rule, with a different index holds, for other kinds of fluc tuating quantities near the c ritic al point (Patashinskii and Pokrovsky 1978). These authors suc c eeded in getting the right general struc ture of the c orrelations. In building upon the early work of Widom, these authors suc c eeded in c onstruc ting most of the elements of the two-index sc aling theory of c ritic al phenomena. Patashinskii and Pokrovsky's work pointed the way toward future field theoretic c alc ulations of c orrelation behavior. In parallel, I c alc ulated (1966) the long-distanc e form of the spin c orrelation func tion for the two-dimensional Ising model by making use of the Onsager solution. This was the part of a long series of c alc ulations that would give insight into the struc ture of that model (Mc Coy and Wu 1973). Those insights would be quite c ruc ial in establishing the fundamental theory of behavior at the c ritic al point. 6.3 Widom Scaling Page 18 of 33

Theories of Matter: Infinities and Renormalization Benjamin Widom (1964; 1965a and b; see figure 4.9) developed a phenomenological theory of the thermodynamic s near c ritic al points. (He studied the liquid– gas transition, but here his results will be stated in the language of the magnetic transition, in whic h the temperature deviation from c ritic ality is t and the symmetry of the ordering is broken by a magnetic field, h.) If t is z ero, the average order parameter, 〈 σ 〉, was experimentally seen to be proportional to h1/δ where δ is a c ritic al index known to be c lose to 4.4 in three dimensions (Widom and Ric e 1955) and 15 in two dimensions (Kadanoff et al. 1967). As disc ussed above, if h = 0 and t 〈 0, then 〈 σ 〉 is proportional to ±(−t)β. He then said that, near the c ritic al point, no one of these three quantities has a natural siz e, but instead eac h one should be measured against the siz e of the others. This led him to suggest (1965) a general formula for the magnetiz ation near the c ritic al point that c ould fit both limiting forms, spec ific ally (17) where g is a function that would have to be determined experimentally. In this way Widom got very concrete and precise results from his initial requirement that each small quantity, 〈 σ 〉, h,t, and so on be measured against another small quantity. He was able to predic t the index-value to desc ribe how every thermodynamic quantity would go to z ero or infinity at c ritic ality. All these c ritic al indic es would then be determined from just the two indices, β and δ. (See Sengers and Shanks (2009) for a comparison of the results of this theory with experiment.) Click to view larger Figure 4.9 Benjamin Widom, left, and Michael Fisher, right. Widom is a Chemistry Professor at Cornell. Fisher has been at King's College (London), Cornell, and the University of Maryland. These results were published in a paper in the Journal of Chemical Physics (1965). In an adjac ent paper, Widom also got a sc aling relation (1965) for the surfac e tension, the free energy of the boundary between liquid and vapor, by relating it to the c oherenc e length. To get this, think of an interfac e c overed by many struc tures produc ed by the c ritic al behavior. One might expec t the c harac teristic siz e and spac ing of these struc tures to be a c orrelation length and that eac h suc h struc ture would bring in an extra free energy of order T. Therefore one expec ts that the entire interfac e would produc e an extra free energy per unit area of order of T times the number of struc tures that c ould be plac ed to fill a unit area, ξ−2 . In view of Eq. (6), whic h asc ribes a c ritic al index ν to ξ, we would have a surfac e tension that varies as t2 ν . This result derived by Widom must have pleased him very muc h, sinc e it shed light on a diffic ulty that went bac k to van der Waals. The theory of the latter gave a c ritic al index for the surfac e tension of 1.5, while van der Waals's and later experiments gave results in the range 1.22 to 1.27. Widom's new theory offered a hope that this old disc repanc y between theory and experiment c ould soon be resolved. An estimation essentially similar to the one Widom used for the surfac e tension indic ates that the singular term in the bulk free energy has an expec ted behavior like t3 ν . This result follows from the idea that the free energy in a three-dimensional system would have its singular part determined by excitations with an energy of order T. These exc itations would have a siz e equal to the c orrelation length and a density equal to the inverse c ube of the c orrelation length. Thus, the surfac e tension and the free energy density provide a bridge between an understanding of the correlation length and an understanding of thermodynamic properties. This bridge is not like anything c ontained in the mean field theories. In fac t these relations, termed hyper-sc aling relations, are the most c harac teristic feature of the renormaliz ation theory that will soon arrive on the sc ene. 6.4 Less Is the Same: Block Transforms and Scaling Page 19 of 33

Theories of Matter: Infinities and Renormalization Widom supplied muc h of the answer to questions about the thermodynamic s of the c ritic al point. I then supplied a part of the strategy for deriving the answer (1966).I will now desc ribe the method in a bit of detail, sinc e the c alc ulation provides some insight into the struc ture of the solution. Imagine c alc ulating the free energy of an Ising model near its c ritic al temperature based upon the interac tions inc orporated in Ising's Hamiltonian func tion for the problem. The result will depend upon the number of lattic e sites, the temperature deviation from c ritic ality and the dimensionless magnetic field. Next imagine redoing the c alc ulation using a new set of variables c onstruc ted by splitting the system into c ells c ontaining several spins and then using new spin variables, eac h intended to summariz e the situation in a bloc k c ontaining several old spin variables. (See figure 4.10.) To make that happen one c an, for example, pic k the new variables to have the same direc tion as the sum of the old spin variables in the bloc k and the same magnitude as eac h of the old variables. The c hange c ould then be represented by saying that the distanc e between nearest neighboring lattic e sites would c hange from its old value, a, to a new and larger value, a′. (See figure 4.10, in whic h the lattic e c onstant has grown by a fac tor ℓ = 3.) In symbols, the c hange is given by (18a) One c an then do an approximate c alc ulation and set up a new “effec tive” free energy c alc ulation that will give the same answer as the old c alc ulation based upon an approximate “effec tive” Hamiltonian making use of the new variables. Near the c ritic al point, one c ould argue on the basis of universality15 that the new Hamiltonian c ould be written to have the same struc ture as the old one. However, near c ritic ality, the new parameters in the effec tive Hamiltonian, the number of lattic e sites, the temperature deviation from c ritic ality, and the dimensionless magnetic field all are proportional to the c orresponding old parameters. This c hange c an be represented by writing (18b) (18c) (18d) In the first of these statements, Eq. (18b), N is the number of lattic e sites and d is the dimension of the lattic e. The equation simply desc ribes how the number of sites depends upon the spac ing between lattic e sites. Figure 4.10 Making blocks. In this illustration a two-dimensional Ising model containing 81 spins is broken into bloc ks, eac h c ontaining 9 spins. Eac h one of those bloc ks is assigned a new spin with a direc tion set by the average of the old ones. We imagine the model is reanalyzed in terms of the new spin variables. The other two equations are far, far less simple. Eq. (18c ) says that the new situation has a symmetry breaking field of the same sign as the previous one. That would be a reflec tion of the fac t that both situations would have the same kind of ordering. The c oeffic ient, (ℓ)yh, might be derived after some sort of statistic al mec hanic al analysis of the situation. It is, as it stands, just a number defined by the result of that c alc ulation and one that might depend upon the exac t way in whic h we c hose to define the new spin variable. Page 20 of 33

Theories of Matter: Infinities and Renormalization The equation for the new value of the new deviation from c ritic ality, t = Kc − K, c ould be desc ribed in similar terms. It is reasonable to assume that if the original system is at its c ritic al point, so is the new desc ription obtained after the block transformation. Further it is reasonable to argue that the transformation should engender no singularities, thus requiring that a new temperature-deviation from c ritic ality would have a linear dependenc e upon the old deviation. So the remaining point is to c alc ulate the c oeffic ient in the linear relation and express it in the spec ial manner given in Eq. (18d). 7. The Wilso n Revo lutio n 7.1 Physical Space; Fourier Space Before entering into Wilson's c onstruc tion of the renormaliz ation group theory, I should touc h upon a point of technique. The proportionalities in Eq. (18d) and Eq. (18c ) are representations of sc aling, and the c oeffic ients in the linear relations define the sc aling relations among the variables. Note that here sc aling is viewed as a c hange in the effec tive values of the thermodynamic parameter produc ed by a c hange in the length sc ale at whic h the system is analyz ed. The length sc ale must be irrelevant to the determination of the eventual answer and must drop out of the final result for the free energy. It is this dropping out that gives the empiric al relations proposed by Widom. These sc aling relations then give a theory with all the empiric al c ontent of Widom's work (1965a), but bac ked by the outlines of a c onc eptual and c alc ulational sc heme. This theoretical work of Kadanoff (1966); Patashinskii and Pokrovsky (1964); and Widom (1965a) was well- rec eived. The review paper of (Kadanoff et al. 1967) was partic ularly aimed at seeing whether the new phenomenology agreed with the experimental data. It reviewed most of the rec ent experiments but missed large numbers of the older ones that are inc luded in Domb (1966) and Levelt– Sengers (1976). All of this ac tivity validated the c onsideration of the c ritic al region as an appropriate subjec t of study and led to a spate of experimental and numeric al work, but hardly any further theoretic al ac c omplishments until the work of Wilson (1971). There are two traditional ways of setting up a Hamiltonian or free energy that will then provide a mic rosc opic desc ription of the system. One way is in c oordinate spac e, the real XYZ spac e in whic h you and I live. This setup is the one we used for the Ising model, the Landau theory, and for the desc ription of the previous subsec tion. It is relatively easy to visualiz e and the most effec tive method for problems in low dimensions, spec ific ally for phase transitions in two dimensions. The other method employs Fourier transforms. It represents every variable in terms of its Fourier transform. For example, the order parameter field of the Landau theory has a transform (19) The integral c overs a spac e of dimensionality d. Using ψ (k) as our basic statistic al variable the Landau free energy may be written as (20) This form in Eq. (20) is used to reac h beyond mean field theory and take into ac c ount possible fluc tuations in the loc al variables that desc ribe the system. To do this, one uses F/T as a kind of a kind of Hamiltonian for phase transition problems. In this use, the k-spac e is divided into small piec es and ψ (k) is taken to be an integration variable in eac h piec e. In this c ontext the expression in Eq. (20) is c alled the Landau-Ginz burg-Wilson free energy. The k-shell integration just desc ribed is easily performed if the free energy inc ludes only linear and quadratic terms in the variable, ψ (k). The fourth-order term provides a problem, one that c an be attac ked by using the 2 Page 21 of 33

Theories of Matter: Infinities and Renormalization renormaliz ation method. The term involving k2 ensures that the c ontribution to the integral for the highest values of k will be small and relatively easily c ontrolled. So, one suc c essively integrates over shells in k-spac e, starting from the highest values of | k| , and working downward. As eac h integral is done, one stops and regroups terms to bring everything bac k c lose to the form of the original Landau-Ginz burg-Wilson free energy. As one does this, the c oeffic ients multiplying the various terms c hange. The k-spac e method is partic ularly appropriate for higher dimensions, going down to roughly three dimensions. It is the usual method of c hoic e in partic le physic s. In statistic al physic s, Wilson and Fisher (1972) have done a very c onvinc ing c alc ulation in whic h they analyz e the behavior near four dimensions by assuming that the fourth-order term is quite small. (See the disc ussion of e-expansion in sec tion 7.3.1 below.) Both real-spac e and k-spac e methods have added c onsiderably to our understanding of phase transitions. I use the former to desc ribe the c onc ept of renormaliz ation, sinc e I find it more natural to think about phenomena in real spac e rather than Fourier spac e. In partic le physic s, however, our basic c onc eptualiz ation is based upon, naturally enough, partic les. These are best followed in k-spac e, sinc e the k labels the momentum of partic les. So the two different formulations are c omplementary, with the best applic ations to problems in different dimensionalities and indeed to different fields of sc ienc e. The extended singularity theorem, of c ourse, applies equally in both the real-spac e and the Fourier-spac e formulations. In real-spac e, in order to have the potential for generating singularities, and thereby phase transitions, the system must be infinite in two or more dimensions. In Fourier-spac e, the c orresponding statement is that two or more c omponents of the k-vec tor must extend to infinity. The remaining requirement in either formulation is that the renormaliz ation must lead to a nontrivial fixed point, one with infinitely large values of some of the couplings. 7.2 Wilson's Contribution Around 1970, these c onc epts were extended and c ombined with previous ideas from partic le physic s (Gell-Mann and Low 1954; Stuec kelberg and Peterman 1953) to produc e a c omplete and beautiful theory of c ritic al point behavior, the renormaliz ation group theory of Kenneth G. Wilson 1971. (See figure 4.11.) The basic idea of reduc ing the number of degrees of freedom, desc ribed in Kadanoff 1966, was extended and c ompleted. Wilson, in essenc e, c onverted a phenomenology into a c alc ulational method by introduc ing ideas not present in the earlier phenomenologic al treatment (Kadanoff 1966): Figure 4.11 Kenneth G. Wilson at California Tech where he did a Ph.D. thesis under Murray Gell-Mann, a major c ontributor to early work on renormalization in partic le physic s. This was followed by a J unior Fellowship at Harvard, a year's stay at CERN, and then an ac ademic appointment at Cornell. The renormalization group work was done while Wilson was at Cornell. • Instead of using a few numbers, for example, t,h, to define the parameters multiplying a few c oupling terms, he extended the list of possible c ouplings to inc lude all the kinds of terms that might be found in the Hamiltonian of the system. Thereby it bec ame automatic ally true that the renormaliz ation would maintain the different c oupling terms, but only c hange the siz e of the parameters whic h multiplied them. Page 22 of 33

Theories of Matter: Infinities and Renormalization • Wilson c onsiders indefinitely repeated transformations, as in the earlier partic le physic s work. Eac h transformation inc reases the siz e of the length sc ale. In c onc ept, then, the transformation would eventually reach out for information about the parts of the system that are infinitely far away. In this way, the infinite spatial extent of the system bec ame part of the c alc ulation. The idea that behaviors at the far reac hes of the system would determine the thermodynamic singularities were thenc e inc luded in the c alc ulation. • Furthermore, Wilson added the new idea that a phase transition would oc c ur when the transformations brought the c oupling to a fixed point. That is, after repeated transformations, the c ouplings all would settle down to a behavior in whic h further renormaliz ation transformation would leave them unc hanged. • Finally, at the fixed point, the c orrelation length would be required to be unc hanged by renormaliz ation transformations. The transformation multiplies the length sc ale by a fac tor that depends upon the details of the transformation. Wilson noted that there are two ways that the c orrelation length might be unc hanged. For transformations related to a c ontinuous transition, the c orrelation length is infinite, thenc e reflec ting the infinite- range c orrelation. For transformations related to first-order transitions, the c orrelation length is z ero, reflec ting the loc al interac tions driving the transition. A very important c orollary to the use of repeated transforms is the idea of running coupling constants. As the length sc ale c hanges, so do the values of the different parameters desc ribing the system. In the earlier field theoretic al work (Gell-Mann and Low 1954; Stuec kelberg and Peterman 1953), the important parameters were the c harge, masses, and c ouplings of the “elementary” partic les desc ribed by the theory. The parameters to be varied were spec ified at the beginning and were, in no sense, the outc ome of the renormaliz ation c alc ulation. The c hange in length sc ale then c hanged these prespec ified parameters from the “bare” values appearing in the basic Hamiltonian to renormaliz ed values that might be observed by experiments examining a larger sc ale. The use of renormaliz ed or “effec tive” c ouplings was c urrent not only in partic le theory but also in the quasipartic le theories that are pervasive in c ondensed matter physic s (Anderson 1997). In these theories one deals with partic les that interac t strongly with one another. Nonetheless, one treats them using the same Hamiltonian formalism that one would use for noninterac ting partic les. The only differenc e from free partic les is that the Hamiltonian is allowed to have a position and momentum dependenc e that reflec ts the c hanges produc ed by the interac tions. In this work, the quantities to be renormaliz ed are prespec ified. In c ontrast, Wilson's renormaliz ation c alc ulation determines what is to be renormaliz ed as a part of the c alc ulation. 7.3 Building upon the Revolution This renormaliz ation theory provided a basis for the development of new methods that c ould be used for building an understanding of c ritic al phenomena and additional subjec ts as diverse as partic le physic s, the development of c haos, the behavior of c omputer programs, as well as dynamic al behavior in c ondensed matter physic s. It provided a framework into whic h one c ould fit a variety of different theories and physic al problems. There was a tremendous flowering of new work following upon Wilson's. 7.3.1 The ε-expansion But first the renormaliz ation method had to gain ac c eptanc e. The most substantial step in that direc tion c ame from the ε-expansion of Wilson and Fisher (1972). Here ε means dimension minus four. This c alc ulational method foc uses upon the dependenc e of physic al quantities upon dimension. It uses renormaliz ation transforms near four dimensions, where mean field theory is almost, but not quite, c orrec t.16 The idea of using the dimension of the system as a c ontinuously variable parameter seems a bit strange at first sight. However, in the momentum-spac e representation of statistic al ensembles, eac h term in a perturbation expansion c an be evaluated for all integer values of the dimension and then the analysis c an be c ontinued to all values of the dimension, inc luding noninteger values. When applied near four dimensions this method allows an almost exac t analysis of the fixed point behavior. Near the fixed point, the nontrivial terms in the free energy, like the term proportional to C in Eq. (20), go to z ero as the dimension approac hes four. Bec ause of this simplific ation, the method gives quite ac c urate results for c ritic al behavior near four dimensions. Further, it provides a series expansion that gives useful answers for many different models in three dimensions. The c lose c orrespondenc e of theory and experiment helped to c onvinc e people that Page 23 of 33

Theories of Matter: Infinities and Renormalization both the variable-dimension method and the renormaliz ation method were valid. The way had been opened for an explosion of new c alc ulations and new understandings. 7.4 Different Kinds of Fixed Points Wilson's theory gives three different kinds of fixed point c orresponding to three qualitatively different points in phase diagrams. For the weak coupling fixed point, c ouplings c an go to z ero and the c orrelation length goes to z ero. The symmetry represented by the order parameter will remain unbroken. This kind of fixed point desc ribes all areas of the phase diagram that do not touc h a phase transition. For the strong coupling fixed point, some c ouplings will go to infinity and the c orrelation length goes to z ero. Here the basic symmetry represented by the order parameter gets broken by at least one nonz ero c oupling that violates that symmetry. This kind of fixed point desc ribes all areas of the phase diagram that touc h a first-order phase transition. For the critical fixed point c ouplings remain finite, the symmetry remains unbroken, and a c orrelation length goes to infinity. Critic al fixed points may be c lassified by their dimension and by the symmetry of their order parameter. The c ombination of the Landau theory and the ε-expansion gave the first steps in that direc tion. The later c alc ulations of c ritic al behavior were then fit into this sc heme. 8. New Co ncepts 8.1 Different Scalings: Relevant, Irrelevant, Marginal Sinc e the Wilsonian point of view generated the renormaliz ation of many different c ouplings, it bec ame important to keep trac k of the different ways in whic h the c ouplings in the free energy would c hange as the length sc ale c hanges. This work starts with an eigenvalue analysis. One takes linear c ombinations of c ouplings and arranges the c ombinations so that, after a renormaliz ation, every c ombination reproduc es itself exc ept for a multiplic ative fac tor. In other words, this approac h makes every linear c ombination of c ouplings obey an equation like the ones in Eq. (18), so that the c ombination, s, obeys (21) The different c ombinations are then c lassified ac c ording to the values of the index, ys, whic h may be c omplex. There are three possibilities (Domb, Green, and Lebowitz 2001; see F. Wegner, Vol. 6, p. 8): • Relevant, real part of ys greater than z ero. These are the c ouplings like t and h that grow larger as the length sc ale is inc reased. Eac h of these will, as they grow, push one away from the c ritic al point. In order to reac h the c ritic al point, one must adjust the initial Hamiltonian so that these quantities are z ero. • Irrelevant, real part of ys less than z ero. These c ouplings will get smaller and smaller as the length sc ale is inc reased so that, as one reac hes the largest length sc ales, they will have effec tively disappeared • Marginal, real part of ys equal to z ero. The last case is rare. Let us put it aside for a moment and argue as if only the first two existed. 8.2 Universality Classes To study c ritic al phenomena based upon renormaliz ation transformations, one sets all the relevant c ombinations of c ouplings to z ero and then does a suffic ient number of suc c essive renormaliz ations so that all the irrelevant c ombinations have effec tively disappeared. We thus end up with a unique fixed point independent of the value of all of the irrelevant c ouplings. The ac t of renormaliz ation is a sort of foc using in whic h many different irrelevant c ouplings fade away and we end up at a single fixed point representing a whole multidimensional c ontinuum of different possible Hamiltonians. These Hamiltonians form what is c alled a universality class. Eac h Hamiltonian in its c lass has exac tly the same c ritic al point behavior, with not only the same c ritic al indic es but also the same long- ranged c orrelation func tions, and the same singular part of the free energy func tion. The identity among different problems is not just a theoretic al artifac t. The Ising model, single axis ferromagnets, and the liquid– gas phase transition all show identic al c ritic al properties (Lee and Yang 1952a and b). The theory Page 24 of 33

Theories of Matter: Infinities and Renormalization makes these c ritic al properties vary with dimension, and experiments bear out the predic ted universality in the two observable c ases: d = 2 and d = 3. As another example, XY ferromagnets have a two c omponent order parameter, with the same symmetry properties as superfluids, with their c omplex order parameter. This universality-c lass idea has been applied to many different problems beyond c ritic al phenomena.17 Whenever two systems show an unexpec ted or deeply rooted identity of behavior they are said to be in the same universality class. There are, of c ourse, many different universality c lasses c orresponding to different dimensionalities, different symmetries of the order parameter, and to different stability properties of the fixed points. Before leaving this subjec t, foc us onc e more on the possibility of a marginal behavior. In the marginal c ase, we have a c oupling that does not vary under renormaliz ation. That kind of c oupling c an produc e c ritic al properties that vary c ontinuously as some parameter is varied. For example, a pair of c oupled Ising models living in the same space show a marginal behavior of this kind (Kadanoff and Wegner 1973). A different marginal behavior is shown by the XY model in two dimensions (Hadz ibabic 2006; Kosterlitz and Thouless 1973). 8.3 New Kinds of Answers In one sense the renormaliz ation group is rather different from anything that had c ome before in statistic al physic s, and by extension in other parts of physic s as well. Previous work in statistic al physic s had emphasiz ed finding the properties of problems defined by statistic al sums, eac h sum being based upon a probability distribution defined by partic ular values of c oupling c onstants like K and h. Suc h sums would be c alled solutions to the problems in question. In the renormaliz ation group work the emphasis is on c onnec ting problems by saying that different problems c ould have identic al solutions. The method involved finding different values of c ouplings that would then give identic al free energies and other properties. These set of c ouplings would then form a representation of a universality c lass. All the interac tions that flow into a given fixed point in the c ourse of an infinite number of renormaliz ations belong to the universality c lass of that fixed point. A universality c lass would give a solution, in the old sense, if one finds within the c lass a set of c ouplings so simple that the solution is obvious. This is what happens when the running c ouplings produc e infinitely weak interac tions, thereby produc ing a weak c oupling fixed point. A strong c oupling fixed point might also be trivial if no important symmetry remains after the order parameter takes on a nonz ero value. However, a first-order phase transition might produc e a nontrivial situation with quite a bit of remaining symmetry. In that c ase, further analysis is nec essary before one c an get anything like a solution in the old sense of the word. Finally, a c ritic al fixed point is not really a “solution” in the old sense. It gives us values of c ritic al indic es and desc ribes sc aling behavior, whic h c an then be used to infer many of the qualitative properties of a solution. But many of the details of the old-sense solution may not be available from a knowledge of the fixed point alone. 8.4 Flows and Flow Diagrams As already mentioned, a renormaliz ation operation differs from the c alc ulations performed within the statistic al mec hanic s of Boltz mann and Gibbs. In statistic al mec hanic s you start with a statistic al ensemble, usually defined with a Hamiltonian, and use that ensemble to c alc ulate an average. In a renormaliz ation operation, you start with a statistic al ensemble, usually defined by a Hamiltonian, and you c alc ulate another ensemble, often desc ribed by a Hamiltonian c ontaining different c ouplings. In one c ase the c alc ulation is, in brief, ensemble generates averages; in the other, the c alc ulation is ensemble generates ensemble. This is quite a substantial differenc e. The part of mathematic s that goes with standard statistic al mec hanic s is probability theory. One part of the mathematic s that goes with renormaliz ation is c alled “dynamic al systems theory” and desc ribes how things c hange under transformations. The c onc epts of a fixed point and of a basin of attrac tion belong to dynamic al systems theory rather than probability theory. Dynamic al systems theory is often used to desc ribe c ontinuous c hanges, as, for example, the c hanges in a mec hanic al system as its state c hanges in time. For the purposes of this sec tion, I will speak as if all renormaliz ation transformations were c ontinuous c hanges produc ed by an infinitesimal inc rease in a basic length. Page 25 of 33

Theories of Matter: Infinities and Renormalization Thus, the transformation will be a → a + dℓ. Then, every c oupling also undergoes an infinitesimal c hange K → K + dK. In partic le physic s this kind of approac h has the name of the Callen-Symanz ik equation (Callan 1970; Symanz ik 1970, 1971). The simplest kinds of flow pic tures look at a single c oupling c onstant, K, and how that c oupling c hanges under renormaliz ation. In the one-dimensional Ising model, depic ted in figure 4.12, eac h renormaliz ation makes the c oupling weaker. Thus the c oupling flows toward the weak c oupling fixed point at K = 0. In c ontrast, the flow in figure 4.13 desc ribes the two-dimensional Ising model. The flow is z ero at the c ritic al fixed point. To the left of the c ritic fixed point, all c ouplings flow toward the weak c oupling fixed point at K = 0; to the right, all flows go toward the strong c oupling point at K = ∞. This diagram desc ribes a system with a single c ritic al point, but a total of three fixed points. Figure 4.12 Flow diagram for one-dimensional Ising model. Renormalization weakens the coupling and pushes it toward a weak c oupling fixed point at K = 0. Figure 4.13 Flow diagram for two-dimensional Ising models. There is a criticial fixed point at K = Kc. For initial c ouplings weaker than this value, renormalizations weaken the c oupling and push it toward a weak c oupling fixed point at K = 0. Conversely, if the initial c ouplings are stronger than Kc, renormalizations produc e a flow toward a strong c oupling fixed point, desc ribing a ferromagnetic state. We have c ome a long way from the starting point set by Boltz mann and Gibbs. Solutions to problems in statistic al mec hanic s have here been desc ribed in terms of renormaliz ation group flows, universality c lasses, and types of fixed points. This new language has bec ome important in statistic al physic s and has been extended to applic ations well beyond the situations desc ribed here. This language, derived from statistic al mec hanic s, has bec ome even more pervasive in partic le physic s, where c oupling c onstants run everywhere. A c alc ulational method is more than a way of putting symbols on paper. It provides a way of looking at, and c onc eptualiz ing, nature. 8.5 The Renormalization Group Is Not a Group Although the renormaliz ation operation is usually desc ribed as a part of a group, bloc k transformations ac tually produc e a semigroup. A group is a set of operations with three c harac teristic s: • Two operations in the group, taken in suc c ession, produc e another group operation. • The group c ontains an element c alled the identity, whic h has the effec t of c hanging nothing whatsoever. • For each operation in the group there is, as part of the group, an inverse operation, so that when you suc c essively perform the operation and its inverse, that pair of operations produc es the identity element. A semigroup lac ks the third c harac teristic . Onc e you have performed a group operation you c annot nec essarily undo that operation. The reason that renormaliz ation produc es a semigroup is that a bloc k transformation (see sec tion 6.4) loses information. After the transformation, the system c ontains fewer lattic e sites and so c an hold less “information.” Some irrelevant c ouplings, whic h c ould be seen before the transformation have simply disappeared. (These c ouplings have the index value y = −∞. In addition, there are other kinds of c ouplings, c alled “redundant,” that do not affec t the free energy and so disappear without a trac e in the c ourse of a renormaliz ation.) Both kinds of c oupling make renormaliz ation a semigroup operation. This c harac teristic is important bec ause it eliminates the possibility of finding the small-sc ale Hamiltonian of the system by looking at large-sc ale phenomena. Before the use of renormaliz ation methods sc ientists often thought that a suffic iently ac c urate and detailed study of a system, albeit a study c onduc ted on a large length sc ale, c ould determine all the basic laws governing the system, down to the smallest sc ale. In prac tic e, the disentanglement of mic rosc opic laws has always proved to be hard. But, in princ iple, it was always assumed to be possible. However, Page 26 of 33

Theories of Matter: Infinities and Renormalization the renormaliz ation group theory says that information will disappear in the proc ess of c hanging length-sc ales. Even ordinary irrelevant operators have effec ts that disappear with exponential rapidity in the c ourse of a renormaliz ation transformation. These do not produc e, in princ iple, a disappearanc e of information but they make it well-nigh impossible to rec onstruc t a small-sc ale Hamiltonian from large-sc ale data. 8.6 A Calculational Method Defines Many Worlds Wilson, in essenc e, c onverted a slightly vague phenomenologic al theory into a well-defined c alc ulational method. So are they all: c lassic al mec hanic s, quantum mec hanic s, statistic al mec hanic s, field theory,.…, all c alc ulational methods. But they are also eac h c omplete desc riptions of some “sub-universe.” As suc h, they eac h engender their own world and their own philosophy. As you c an see from the set of ideas outlined in this sec tion, the renormaliz ation group built its own set of philosophic al perspec tives, whic h then displayed statistic al physic s and c ondensed matter physic s in a new way. However, there is an additional sense in whic h the renormaliz ation group defines its own worlds. Eac h fixed point has its own basin of attrac tion defined by its very large set of irrelevant c ouplings. This basin of attrac tion is the region in the spac e of possible Hamiltonians that will eventually flow into our partic ular fixed point. Within this basin, the flow of the sc aling variables play a c ruc ial role. When two or more of these variables interfere, we c an see non- linear effec ts. Sinc e we expec t that our listing of variables is a c omplete list, we expec t that two variables ac ting together will produc e an effec t that we c an desc ribe as a summed effec t of the variables on our previous list. In this way, we get a kind of multiplic ation table in whic h the produc t of any two variables is a sum of the others with spec ified c oeffic ients. Suc h a multiplic ation table is what the mathematic ians c all an algebra. This algebra defines what is happening in the phase transition. The algebras that have ac tually been studied are a little deeper than the one just desc ribed. They are produc ed not just by the c ouplings, but by the spec ific ations of the c ouplings in loc al regions of the system. Therefore the algebras c ombine the properties of spac e with the properties of the partic ular fixed point. They have been most ric hly studied in two dimensions (Belavin, Polyakov, and Zamolodc hikov 1984; Friedan, Qiu, and Shenker 1984) in whic h the spatial part c ommon to all these algebras is c alled the Virasoro algebra (Virasoro 1970). This approac h also plays an important role in string theory. Eac h fixed point has its own unique algebra (Kadanoff 1969; Wilson 1969), c alled a short distanc e expansion or an operator produc t expansion, that desc ribes the struc ture of the loc al c orrelations determining the fixed point behavior. 8.7 Extended Singularities Revisited The renormaliz ation group has an entirely different spatial struc ture from that of mean field theory. The differenc e c an best be seen by c omparing the Ising model mean field theory of Sec tion (3) with the bloc k spin formulation of Section (6.4). In the mean field formulation, the value of an average magnetiz ation at point r is determined, first of all, by the values of the magnetiz ation at points c onnec ted by bonds to the initial point. These are then determined, in turn, by magnetiz ations at points c onnec ted to these new points by bonds. This extension proc ess might c ontinue indefinitely or terminate after only finding a finite number of spins. In either c ase, the theory may or may not predic t a phase transition. Mean field theory does not have the right spatial struc ture for the c orrec t predic tion of phase transitions. In c ontrast, the bloc king proc edure of the renormaliz ation group determines the c ouplings in a given region, in the first analysis, by the effec ts of c ouplings in a region of siz e ℓ larger. The bloc king then reac hes out in geometric progression to regions eac h expanded by a fac tor of ℓ. Of c ourse, the bloc k transformation reac hes out more quic kly and effec tively than do the steps of the mean field c alc ulation. But that is not the main differenc e. The mean field theory c an have a pseudo-phase transition determined by just a few c ouplings. On the other hand, if the bloc k transformation ever reac hes out and sees no more c ouplings in the usual approximation sc hemes (Niemeijer and Leeuwen 1973) that will signal the system that a weak c oupling situation has been enc ountered and will c asc ade bac k to produc e a weak c oupling phase. Henc e Page 27 of 33

Theories of Matter: Infinities and Renormalization the bloc king approac h has the potential of using the right fac t about the spatial topology to determine the possibility of a phase transition. By this argument the extended singularity theorem suggests that phase transitions are triggered by a very elegant mathematic al juxtaposition put before us by Nature. On one hand, the phase transition is c onnec ted with a symmetry operation built into the mic rosc opic c ouplings of the system. For example, the ferromagnetic based upon the breaking of a symmetry in the possible direc tion of spins. On the other hand, the phase transitions also make use of the extended topology of a system that extends over an effec tively infinite region of spac e. This c oupling of mic rosc opic with mac rosc opic has an unexpec ted and quite breathtaking beauty. Some of the material in this review was first prepared for a talk I gave at the Royal Netherlands Ac ademy of Arts and Sciences in 2006. Still more of the material appeared in a talk at the 2009 Seven Pines meeting on the Philosophy of Physic s under the title “More Is the Same, Less Is the Same, Too; Mean Field Theories and Renormaliz ation.” These talks have appeared on the authors’ website (2009) sinc e then. This Seven Pines meeting was generously sponsored by Lee Gohlike. The paper also inc orporates material from “More Is the Same” published in J. Stat. Phys. in 2009. This work was supported in part by the University of Chic ago MRSEC program under NSF grant number DMR0213745. It was c ompleted during visits to the Perimeter Institute, whic h is supported by the Government of Canada through Industry Canada and by the Provinc e of Ontario through the Ministry of Researc h and Innovation, and by the present NSF DMR-MRSEC grant number 0820054. I had useful disc ussions related to this essay with Tom Witten, E. G. D. Cohen, Gloria Lubkin, Ilya Gruz berg, Gene Maz enko, Hans van Leeuwen, Wendy Zhang, Franz Wegner, Roy Glauber, Yitz hak Rabin, Gerard't Hooft, Sidney Nagel, and Subir Sac hdev. I owe partic ular thanks to Mic hael Fisher who, as he has done many times, helped me understand this interesting subjec t. References H. C. Andersen, J. D. Weeks, and D. Chandler. Relationship between the hard-sphere fluid and fluids with realistic repulsive forces. Phys. Rev. A, 4:1597–1607, 1917. P. W. Anderson. More is different. Science, 177:393–396, 1972. P. W. Anderson. Concepts in Solids. World Sc ientific , Singapore, 1997. A. F. Andreev. Singularity of thermodynamic quantities at a first order phase transition point (Singularity of thermodynamic potential for liquid at boiling point). Sov. Phys. JETP, 18:1415, 1964. T. Andrews. On the c ontinuity of the gaseous and liquid states of matter. Phil. Trans. Roy. Soc., 159: 575– 590, 1869. M. I. Bagatskii, A. V. Voronel’, and V. G. Gusak. Measurement of the spec ific heat Cv of Argon in the immediate vic inity of the c ritic al point. JETP, 16: 517, 1963. J. Bardeen, L. N. Cooper, and J. R. Sc hrieffer. Mic rosc opic theory of superc onduc tivity. Phys. Rev., 106: 162– 164, 1957. J. Bardeen, L. N. Cooper, and J. R. Sc hrieffer. Theory of superc onduc tivity. Phys. Rev., 108: 1175, 1957. J. G. Bednorz and K. A. Müller. Possible high Tc superc onduc tivity in the Ba-La-Cu-O system. Z. Physik, B 64: 189– 193, 1986. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodc hikov. Infinite c onformal symmetry in two-dimensional quantum field theory. Nucl. Phys., B241:333–380, 1984. Stephen G. Brush. History of the Lenz -Ising model. Rev. Mod. Phys, 39, 1967. Stephen G. Brush. Statistical Physics and the Atomic Theory of Matter from Boyle and Newton to Landau and Page 28 of 33

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Theories of Matter: Infinities and Renormalization 1965. United States Department of Commerc e, National Bureau of Standards, Washington, 1965. Th. Niemeijer and J. M. van Leeuwen. Wilson Theory for Spin Systems on a Triangular Lattic e. Phys. Rev. Letts., 31:1411, 1973. Martin Niss. Phenomena, Models, and Understanding, The Lenz-Ising Model and Critical Phenomena 1920–1971. Springer, Berlin/Heidelberg, 2005. Nai-Phuan Ong and Ravin Bhatt, editors. More Is Different: Fifty Years of Condensed Matter Physics. Princ eton Series in Physic s, Princ eton, NJ, 2001. L. Onsager. Crystal Statistic s. I. A Two-Dimensional Model with an Order-Disorder Transition Phys. Rev., 65: 117, 1944. L. S. Ornstein and F. Zernike. Ac c idental deviations of density and opalesc enc e at the c ritic al point of a single substance. Proc. Acad. Sci. Amsterdam, 17:793, 1914. Abraham Pais. Subtle Is the Lord … Oxford University Press, paperbac k, 1983. A. Z. Patashinskii and V. L. Pokrovsky. Sec ond order phase transition in a Bose liquid. Soviet Phys. JETP, 19: 667, 1964. A. Z. Patashinskii and V. L. Pokrovsky. Fluctuation Theory of Phase Transitions. Amsterdam, Elsevier, 1978. Muriel Rukeyser. Willard Gibbs: American Genius. Ox Bow Press, Woodbridge, CT, 1942. Jan V. Sengers and Joseph G. Shanks. Experimental c ritic al-exponent values for fluids. J. Stat. Phys., 137: 857– 877, 2009. E. C. G. Stuec kelberg and A. Peterman. Normaliz ation of c onstants in the quantum theory. Helv. Phys. Acta, 26:499, 1953. K. Symanz ik. Small-distanc e behaviour in field theory and power c ounting. Commun. Math. Phys., 18: 227, 1970. K. Symanz ik. Small-distanc e-behaviour analysis and Wilson expansions. Commun. Math. Phys., 23: 49, 1971. G. ’t Hooft. Renormaliz able Lagrangians for massive Yang-Mills fields. Nucl. Phys. B, 35: 167, 1971. G. ’t Hooft and M. Veltman. Regulariz ation and renormaliz atioin of gauge fields. Nucl. Phys. B, 44: 189, 1972. G. ’t Hooft and M. Veltman. Combinatoric s of gauge fields Nucl. Phys. B, 50: 318, 1972. Jos Uffink. Rereading Ludwig Boltz mann. In L. Valdes and D. Westerstahl, editors, Methodology and Philosophy of Science. Proceedings of the Twelfth International Congress, pages 537– 555. King's College Public ations, London, 2005. George E. Uhlenbec k. Some historic al and c ritic al remarks about the theory of phase transitions. In Shigeji Fujita, editor, Science of Matter. Festschrift in Honor of Professor Ta-You Wu. Gordon and Breac h Sc ienc e Publishers, New York, 1978. J. D. van der Waals. Over de Continuiteit van den Gas- en Vloeistoftoestand, Leiden, The Netherlands, 1873. J. E. Verschaffelt. Commun. Phys. Lab. Leiden, (55), 1900. M. A. Virasoro. Subsidiary c onditions and ghosts in dual-resonanc e models. Phys. Rev, D1: 2933– 2936, 1970. A. V. Voronel’, Yu. Chaskin, V. Popov, and V. G. Simpkin. JETP, 18:568, 1964. P. Weiss. J. Phys., 6:661, 1907. B. Widom. J. Chem. Phys. 41:1633, 1964 Page 31 of 33

Theories of Matter: Infinities and Renormalization B. Widom. Surfac e Tension and Molec ular Correlations near the Critic al Point. J. Chem. Phys., 43: 3892, 1965a. B. Widom. Equation of State in the Neighborhood of the Critic al Point. J. Chem. Phys., 43: 3898, 1965b. B. Widom and O. K. Ric e. Critic al Isotherm and the Equation of State of Liquid-Vapor Systems. J. Chem. Phys., 23:1250, 1955. K. Wilson. Non-Lagrangian Models of Current Algebra Phys. Rev., 179:1499, 1969. K. G. Wilson. Renormaliz ation group and c ritic al phenomena. I. Renormaliz ation group and the Kadanoff sc aling picture. Phys. Rev., B4:3174–3183, 1971. K. G. Wilson and M. E. Fisher. Critic al exponents in 3.99 dimensions. Phys. Rev. Lett., 28: 240– 243, 1972. C. N. Yang. The Spontaneous Magnetiz ation of a Two-Dimensional Ising Model. Phys. Rev., 85: 808, 1952. Notes: (1) The word “phase” is interesting. Ac c ording to the Oxford Dictionary of Word Histories (and the Oxford English Dictionary) it entered English language in the nineteenth c entury to desc ribes the phases of the moon. The Oxford English Dictionary lists a very early use in J. Willard Gibbs's writings about thermodynamic s as the “phases of matter.” Apparently Gibbs then extended the meaning to get “extension in phase” that then got further extended into the modern usages “phase transition” and “phase spac e.” (2) Partic le physic s does show a very weak time reversal asymmetry disc overed by James W. Cronin and Val L. Fitch, but this asymmetry is immaterial for all mundane phenomena. (3) This func tion is named after William Rowan Hamilton who desc ribed how to formulate c lassic al mec hanic s using this Hamiltonian func tion. (4) The word “c anonic al” seems to be a somewhat old-fashioned usage for something set to a given order or rule. The Oxford English Dictionary trac es it bac k to Chauc er. (5) I use energy units in order to write fewer symbols. It is more c onventional to write, instead of T, kT, where k is the Boltz mann c onstant. (6) In fac t, the liquid– gas c ase is one of the most subtle of the phase transitions sinc e the symmetry between the two phases, gas and liquid, is only an approximate one. In magnets and most other c ases the symmetry is essentially exact, before is it broken by the phase transition. (7) As pointed out to me by Hans van Leeuwen, the opalesc enc e is very c onsiderably enhanc ed by the diffic ulty of bringing the near-c ritic al system to equilibrium. The out-of-equilibrium system tends to have anomalously large droplets analogous to those produc ed by boiling. These droplets then produc e the observed turbidity. (8) Einstein then used the explanation of this physic al effec t to provide one of his several suggested ways of measuring Avogadro's number, the number of molec ules in a mole of material. (9) Muc h of the historic al material in this work is taken from the exc ellent book on c ritic al phenomena by Cyril Domb (1966). (10) Imprec ision c an often be used to distinguish between the mathematic ian and the physic ist. The former tries to be prec ise; the latter sometimes uses vague statements that c an then be extended to c over more c ases. However, in precisely defined situations, for example the situation defined by the Ising model, the extended similarity “theorem” is actually a theorem (Isakov 1984). (11) Calc ulations of the effec ts of sc ale c hanges are muc h more implic it in mean field theories than in renormaliz ation theories. In both c ases we are treating variation over a huge range of sc ales, and power laws are a likely way of desc ribing this huge range of variation. However, bec ause the mean field theories deal less direc tly with sc ale transformations they do not get the relation between the renormaliz ation sc alings of fluc tuations and free Page 32 of 33

Theories of Matter: Infinities and Renormalization energy quite right. (12) The symmetry of the phase transition is reflec ted in the nature of the order parameter, whether it be a simple number (the c ase disc ussed here), a c omplex number (superc onduc tivity and superfluidity), a vec tor (magnetism), or something else. (13) Onsager's results looked different from those depic ted in figure 4.8 in that they showed muc h more symmetry between the high-temperature region and the low-temperature region. This differenc e reflec ts the fac t that two- dimensional c ritic al phenomena are markedly different in detail from three-dimensional c ritic al phenomena. Further, subsequent work has indic ated that none of the heat c apac ity singularities shown in figure 4.8 are ac tually logarithmic in c harac ter. They are all power law singularities. (14) There are exc eptions. Mean field theory works quite well whenever the forc es are suffic iently long-ranged so that many different partic les will interac t direc tly with any given partic le. By this c riterion mean field theory works well for the usual superc onduc ting materials studied up through the 1980s(7, 8), exc ept extremely c lose to the c ritic al point. However, mean field theory does not work for the newer “high-temperature superc onduc tors,” a c lass disc overed in 1986 by Georg Bednorz and Alexander Müller(9). (15) I used Eq. (18), but I did NOT make an explic it argument based upon universality in my paper in whic h I first applied this bloc k transformation. My disc ussion would have been muc h stronger had I the wisdom to do so. But wisdom often c omes after the fac t. (16) The idea of variable dimensionality is also used in partic le physic s under the name dimensional regulariz ation. One of the earliest applic ations in partic le physic s was in the work of't Hooft and Veltman (1972a and b) proving that the gauge theory of strong interac tions was renormaliz able. (17) I must admit to a c ertain pride c onnec ted with universality. The 1967 review paper (Kadanoff et al.) in whic h I partic ipated was organiz ed about universality c lasses. I borrowed the word “universality” from the c onversation of Sasha Polyakov and Sasha Migdal, who were apparently translating a usage c ommon in the Landau group. I then imported this usage into the English language (1990). Alternatively, one might argue that universality was a produc t of many different authors, inc luding Robert Griffiths (1975), as well as the entire King's College sc hool (Domb 1996). Leo P. Kad anoff Leo P. Kadanoff is a theoretical physicist and applied m athem atician who has contributed widely to research in the properties of m atter, the developm ent of urban areas, statistical m odels of physical system s, and the developm ent of chaos in sim ple m echanical and fluid system s. His best-known contribution was in the developm ent of the concepts of \"scale invariance\" and \"universality\" as they are applied to phase transitions. More recently, he has been involved in the understanding of singularities in fluid flow.

Turn and Face The Strange … Ch-Ch-Changes: Philosophical Questions Raised by Phase Transitions Tarun Menon and Craig Callender The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This chapter addresses the question of whether phase transitions are to be understood as genuinely emergent phenomena, disc ussing c onc epts invoked in the inc reasing number of public ations on emergenc e and phase transitions and the c onc eptions of reduc tion and c orresponding notions of emergenc e. It also c onsiders rec ent attempts to provide well-defined notions of phase transition for finite systems and highlights the evolving nature of our philosophic al understanding of phase transitions, emergenc e, and reduc tionism. K ey words: ph ase tran si ti on s, emergen t ph en omen a, fi n i te sy stems, emergen ce, redu cti on i sm, ph i l osoph i cal u n derstan di n g Phase transitions are abrupt c hanges in the mac rosc opic properties of a system. Examples of the phenomenon are familiar: freez ing, c ondensation, magnetiz ation. Often these transitions are partic ularly dramatic , as when solid objec ts c omposed of the silvery metal gallium vanish into puddles when pic ked up (the temperature of the hand is just enough to raise gallium's temperature past its melting point). Charac teriz ed generally, one finds them inside and outside of physic s, in systems as diverse as neutron stars, DNA helic es, financ ial markets, and traffic . In the past half-c entury, the study of phase transitions and c ritic al phenomena has been a c entral preoc c upation of the statistic al physic s c ommunity. In fac t, it is now a truly interdisc iplinary area of researc h. Phase transitions manifest at many different sc ales and in all sorts of systems, so they are of interest to atomic physic ists, materials engineers, astronomers, biologists, soc iologists, and ec onomists. However, philosophic al attention to the foundational issues involved has thus far been limited. This is unfortunate bec ause the theory of phase transitions is unusual in many ways and offers a novel perspec tive that c ould enric h a number of debates in the philosophy of sc ienc e. In partic ular, questions about reduc tion, emergenc e, explanation, and approximation all arise in a partic ularly stark manner when c onsidering this phenomenon. Here we will foc us on these questions as they relate to the most studied type of phase transition, namely, transitions between different equilibrium phases in thermodynamic s. These are sudden c hanges between one stable thermo-dynamic state of matter and another while one smoothly varies a parameter. A paradigmatic example is the c hange in water from liquid to gas as the temperature is raised or the pressure is reduc ed. In the small philosophic al c ommentary on this topic , suc h c hanges have provoked many surprising c laims. Many have c laimed that phase transitions c annot be reduc ed to statistic al mec hanic s, that they are truly emergent phenomena. The argument for this c onc lusion hangs on one's understanding of the infinite idealiz ation invoked in the statistic al mec hanic al treatment of phase transitions. In this c hapter we will foc us on puz z les assoc iated with this idealiz ation. Is infinite idealiz ation nec essary for the explanation of phase transitions? If so, does it show that phase transitions are, in some sense, emergent phenomena? If so, what prec isely is that sense? Questions of this sort provide a c onc rete basis for the exploration of philosophic al approac hes to reduc tion and idealiz ation, and they also bear on the ongoing sc ientific study of these systems. Page 1 of 24

Turn and Face The Strange … Ch-Ch-Changes 1. The Physics o f Phase Transitio ns Phase transitions raise interesting questions about intertheoretic relationships bec ause they are studied from three distinc t theoretic al perspec tives. Thermodynamic s provides a mac rosc opic , phenomenologic al c harac teriz ation of the phenomenon. Statistic al mec hanic s attempts to ground the thermodynamic treatment by explaining how this mac rosc opic behavior arises out of the interac tion of mic rosc opic degrees of freedom. This projec t has led to the employment of renormaliz ation group theory, a tool first developed in the c ontext of partic le physic s for studying the behavior of systems under transformations of sc ale. While renormaliz ation group theory is usually plac ed under the broad rubric of statistic al mec hanic s, the methods employed are importantly different from the traditional tools of statistic al mec hanic s. Rather than a probability distribution over an ensemble of c onfigurations of a single system, the primary theoretic al devic e of renormaliz ation group theory is the flow generated by the sc aling transformation on a spac e of Hamiltonians representing distinc t physic al systems. In this sec tion we desc ribe how these three approac hes treat the phenomenon of phase transitions, with spec ial attention to the employment of the infinite partic le idealiz ation. 1.1 Thermodynamic Treatment The thermodynamic treatment of phases and phase transitions began in the nineteenth c entury. Experiments by Andrews, Clausius, Clapeyron, and many others provided data that would lead to developed theories of phase transitions and c ritic al phenomena. Gradually it was rec ogniz ed that at c ertain values of temperature and pressure a substanc e c an exist in more than one thermodynamic phase (e.g., solid, liquid), while at other values there c an be a c hange in phase but no c oexistenc e of phases. For instanc e, as pressure is reduc ed or temperature is raised, liquid water transitions to its gaseous phase. At the boundary between these phases, both liquid and gaseous states c an c oexist; the thermodynamic parameters of the system do not pic k out a unique equilibrium phase. In fac t, at the triple point of water (temperature 273.16K and pressure 611.73 Pa), all three phases—solid, liquid, and gas—c an c oexist. The transitions at these phase boundaries are marked by a disc ontinuity in the density of water. As the pressure is reduc ed at a fixed temperature, the equilibrium state of water switc hes abruptly from a high-density liquid phase to a low-density gaseous phase. This is an example of a first-order phase transition. As the temperature is inc reased past the c ritic al temperature of 647 K, water enters a new phase. In this regime, there are no longer mac rosc opic ally distinc t liquid and gas phases, but a homogenous superc ritic al fluid that exhibits properties assoc iated with both liquids and gasses. Changing the pressure leads to a c ontinuous c hange in the density of the fluid; there are no phase boundaries. This superc ritic al phase allows a transition from liquid to gas that does not involve any disc ontinuity in thermodynamic observables: raise the temperature of the liquid past the c ritic al temperature, reduc e the pressure below the c ritic al pressure (22 MPa for water), then c ool the fluid bac k to below the c ritic al temperature. This path takes the system from liquid to gas without c rossing a phase boundary. The transition of a system past its c ritic al point to the superc ritic al phase is a continuous phase transition. Mathematic ally, phase transitions are represented by nonanalytic ities or singularities in a thermodynamic potential. A singularity is a point at whic h the potential is not infinitely differentiable, so at a phase transition some derivative of the thermo-dynamic potential c hanges disc ontinuously. A c lassific ation sc heme due to Ehrenfest provides the resourc es to distinguish between first- and sec ond-order transitions in this formalism. A first-order phase transition involves a disc ontinuity in the first derivative of a thermodynamic potential. In the liquid– gas first-order transition, the volume of the system, a first derivative of the thermodynamic potential known as the Gibbs free energy, c hanges disc ontinuously. For a sec ond-order phase transition the first derivatives of the potentials are c ontinuous, but there is a disc ontinuity in a sec ond derivative of a thermodynamic potential. At the liquid– gas c ritic al point, we see a disc ontinuity in the c ompressibility of the fluid, whic h is a first derivative of volume and henc e a sec ond derivative of the Gibbs free energy. Ehrenfest's sc heme extends naturally to allow for higher-order phase transitions as well. An n-th order transition would be one whose n-th derivative is disc ontinuous. Contemporary statistic al mec hanic s retains the c ategory of first-order phase transitions (sometimes referred to as abrupt transitions), but all other types of non-analytic ities in thermodynamic potentials are grouped together as c ontinuous phase transitions. Continuous phase transitions are often referred to as order–disorder transitions. There is usually some symmetry in the superc ritic al phase that is broken when we c ross below the c ritic al point. This broken symmetry allows for the Page 2 of 24

Turn and Face The Strange … Ch-Ch-Changes material to be ordered in various ways, c orresponding to different phases. A stark example of the transition between order and disorder is the transition in magnetic materials, suc h as iron, between paramagnetism and ferromagnetism. At room temperature, a piec e of iron is permanently magnetiz ed when exposed to an external magnetic field. In the presenc e of a field, the minimum energy c onfiguration is the one with the largest possible net magnetic moment reinforc ing the field, so the individual dipoles within the iron align to maximiz e the net moment. This configuration remains stable even when the external field is removed. Materials with this propensity for induc ed permanent magnetiz ation are c alled ferromagnetic. If the temperature is raised above 1043 K, the ferromagnetic properties of iron vanish. The iron is now paramagnetic; it c an no longer sustain induc ed magnetiz ation when the external field is removed. In the stable c onfiguration, there is no c orrelation between the alignments of neighboring dipoles. In the paramagnetic phase, no direc tion is pic ked out as spec ial after the magnetic field is switc hed off. The material exhibits spatial symmetry. In the ferromagnetic phase, this symmetry is broken. The dipoles line up in a partic ular spatial direc tion even after the field is removed. The order represented by this alignment does not survive the transition past c ritic ality. A simple way to understand this transition between order and disorder is in terms of the minimiz ation of the Helmholtz free energy of the system: (1) Here E is the energy of the system, T is the temperature, and S is the entropy. The stable c onfiguration minimiz es free energy. At low temperatures, the energy term dominates, and the low-energy c onfiguration with dipoles aligned is favored. At high temperatures, the entropy term dominates, and we get the high-entropy c onfiguration with unc orrelated dipole moments. The c hange in magnetic behavior is explic able as a shift in the balanc e of power in the battle between the ordering tendenc y due to minimiz ation of energy and the disordering tendenc y due to maximiz ation of entropy. As indic ated, the paramagnetic –ferromagnetic transition is c ontinuous, not first order. All first derivatives of the free energy are c ontinuous, but sec ond derivatives (suc h as the magnetic susc eptibility χ = ∂2F , where H is the magnetiz ation) are not. ∂H 2 The transition from order to disorder is also represented, following Landau, as the vanishing of an order parameter. In the c ase under c onsideration, this parameter is the net magnetiz ation M of the system. Below the c ritic al point, you have different phases with distinc t values of the order parameter. If we simplify our model of the magnetic material so that the induc ed magnetiz ation of the dipoles is only along one spatial axis (as in the Ising model), then at eac h temperature below c ritic ality the order parameter c an take two values, related by a c hange of sign. The magnetiz ation vanishes as we approac h the c ritic al point and remains z ero in the superc ritic al phase, c orresponding to a disappearanc e of distinc t phases. The vanishing of the order parameter c lose to the c ritic al temperature Tc is c harac teriz ed by a power law: (2) where t is the reduc ed temperature (T — Tc)/Tc. The exponent β c harac teriz es the rate at whic h the magnetiz ation falls off as the c ritic al temperature is approac hed. It is an example of a critical exponent, one of many that appear in power laws c lose to the c ritic al point. The experimental and theoretic al study of c ritic al exponents has been crucial to recent developments in the theory of phase transitions. 1.2 Statistical Mechanical Treatment Statistic al mec hanic s is the theory that applies probability theory to the mic rosc opic degrees of freedom of a system in order to explain its mac rosc opic behavior. The tools of statistic al mec hanic s have been extremely suc c essful in explaining a number of thermodynamic phenomena, but it turned out to be partic ularly diffic ult to apply the theory to the study of phase transitions. There were two signific ant obstac les to the development of a suc c essful statistic al mec hanic al treatment of phase transitions: one experimental and one c onc eptual. The experimental obstac le had to do with the failure of mean field theory. This was the dominant approac h to the statistic al mec hanic s of phase transitions up to the middle of the twentieth c entury. The theory is best explic ated by c onsidering the Ising model, whic h represents a system as a lattic e of sites, eac h of whic h c an be in two different states. The states will be referred to as spin up and spin down, in analogy with magnetic systems. However, Ising models have been suc c essfully applied to a number of different systems, inc luding the liquid-gas Page 3 of 24

Turn and Face The Strange … Ch-Ch-Changes system near its c ritic al point. The Hamiltonian for the Ising model involves a c ontribution by an external term, c orresponding to the external magnetic field for magnetic systems, and internal c oupling terms. The only c oupling is between neighboring spins on the lattic e. It is energetic ally favorable for neighboring spins to align with one another and with the external field. This model is supposed to represent the way in whic h loc al interac tions c an produc e the kinds of long-range c orrelations that c harac teriz e a thermodynamic phase. In statistic al mec hanic s, all thermodynamic func tions are determined by the c anonic al partition func tion. The c oupling terms in the Hamiltonian make the c alc ulation of the partition func tion for the Ising model mathematic ally diffic ult. To make this c alc ulation trac table, we approximate the c ontribution of a partic ular lattic e site to the energy of the system by supposing that all its neighbors have a spin equal to the ensemble average. This approximation ignores fluc tuations of spins from their mean values. The fluc tuations bec ome less relevant as the number of neighbors of a partic ular lattic e site inc reases, so the mean field approximation works better the higher the dimensionality of the system under c onsideration. Onc e the partition func tion is c alc ulated using this approximation, there is an elegant method due to Landau for determining the c ritic al exponents. Unfortunately, Landau's method gives results that c onflic t with experiment. For instanc e, the mean field value for the c ritic al exponent β is 0.5, but observation suggests the ac tual value is about 0.32. The approximation fails c lose to the c ritic al point of a magnetic system. In fac t, this failure is predic ted by Landau theory itself. The theory tells us that as we approac h the c ritic al point, the correlation length diverges. This is the typic al distanc e over whic h fluc tuations in the mic rosc opic degrees of freedom are c orrelated. As this length sc ale inc reases, fluc tuations bec ome more relevant, and the mean field approximation, whic h ignores fluc tuations, weakens. Mean field theory c annot fully desc ribe c ontinuous phase transitions bec ause of this failure near c ritic ality Another approac h is needed for a full statistic al mec hanic al treatment of the phenomenon. As mentioned, there was also a deeper c onc eptual obstac le to a statistic al mec hanic s of phase transitions. If one adopts the definition of phase transitions employed by thermodynamic s, then phase transitions in statistical mechanics do not seem possible. The impossibility c laim c an be explained very easily. As mentioned above, thermodynamic func tions are determined by the partition func tion. For instanc e, the Helmholtz free energy is given by: (3) where k is Boltz mann's c onstant, T is the temperature of the system, and Z is the c anonic al partition func tion: (4) with En labeling the different possible mec hanic al energies of the system. Rec all the definition of a phase transition ac c ording to thermodynamic s: (Def 1) An equilibrium phase transition is a nonanalytic ity in the free energy. Depending on the c ontext, one might c hoose a nonanalytic ity in a different thermodynamic potential; however, that freedom will not affec t matters here. As natural as it is, Def 1 makes a phase transition seem unattainable in statistic al mec hanic s. The reason is that eac h of the exponential func tions in (4) is analytic , the partition func tion is just a sum of exponentials, and the free energy essentially is just the logarithm of this sum. Sinc e a sum of analytic func tions is itself analytic and the logarithm of an analytic func tion itself analytic , the Helmholtz free energy, expressed in terms of the logarithm of the partition func tion, will also be analytic . Henc e, there will be no phase transitions as defined by Def 1. Sinc e the same reasoning c an be applied to any thermodynamic func tion that is an analytic func tion of the c anonic al partition func tion modific ations of Def 1 to other thermodynamic func tions will not work either. (For a rigorous proof of the above claims, see Griffiths (1972).) In the standard lore of the field, this problem was resolved when Onsager in 1944 demonstrated for the first time the existenc e of a phase transition from nothing but the partition func tion. He did this rigorously for the two-dimensional Ising model with no external magnetic field. How did Onsager manage the impossible? He worked in the thermodynamic limit of the system. This is a limit where the number of partic les in the system N and the volume of the system V go to infinity while the density ρ = N/V is held fixed. Letting N go to infinity is the c ruc ial tric k in getting around the “impossibility” c laim. The c laim depends on the sum of exponentials in (4) being finite. Any finite sum of analytic func tions will be analytic . Onc e this restric tion is removed, however, it is possible to find nonanalytic ities in the free energy. The apparent lesson is that statistic al mec hanic s c an desc ribe phase transitions, but only in Page 4 of 24

Turn and Face The Strange … Ch-Ch-Changes infinite particle systems. It is c ommon to visualiz e what is going on in terms of the Yang-Lee theorem. The free energy is a logarithm of the partition func tion, so it will exhibit a singularity where the partition func tion goes to z ero. But the partition func tion is a polynomial of finite degree with all positive c oeffic ients, so it has no real positive roots. Instead the roots are imaginary and the z eros of the partition func tion must be plotted on the c omplex plane. The Yang-Lee theorem, for a two-dimensional Ising model, says that these z eros sit on the unit sphere in the c omplex plane. As the number of partic les inc reases, the z eros bec ome denser on the unit sphere until at the thermodynamic limit they intersec t the positive real axis. Sinc e a real z ero of the partition func tion is only possible in this limit, it is only in this limit that we can have a phase transition (understood as in Def 1). An alternative definition of phase transitions is sometimes used, one proposed by Lebowitz (1999). A phase transition oc c urs, on this definition, just in c ase the Gibbs measure (a generaliz ation of the c anonic al ensemble) is nonunique for the system. This c orresponds to a c oexistenc e of distinc t phases and therefore a phase transition. Using this alternative definition, however, will not c hange philosophic al matters. The Gibbs measure c an only be nonunique in the thermodynamic limit, just as Def 1 c an only be satisfied in the thermodynamic limit. That said, this way of looking at the issue perhaps makes it easier to see the similarities between the foundational issues raised by phase transitions and those raised by spontaneous symmetry breaking. 1.3 Renormalization Group Theory We mentioned in the previous sec tion that mean field theory fails near the c ritic al point for c ertain systems bec ause it neglec ts the importanc e of fluc tuations in this regime. Dealing with this strongly c orrelated regime required the introduc tion of a new method of analysis, imported from partic le physic s. This is the renormaliz ation group method. While mean field theory hews to tools and forms of explanation that are orthodox in statistic al mec hanic s, suc h as determining aggregate behavior by taking ensemble averages, renormaliz ation group theory introduc ed a somewhat alien approac h with tools more akin to those of dynamic al systems theory than statistic al mechanics. To explain the method, we return to our stalwart Ising model. Suppose we c oarse-grain a 2-D Ising model by replac ing 3 × 3 bloc ks of spins with a single spin pointing in the same direc tion as the majority in the original bloc k. This gives us a new Ising system with a longer distanc e between lattic e sites, and possibly a different c oupling strength. You c ould look at this c oarse-graining proc edure as a transformation in the Hamiltonian desc ribing the system. Sinc e the Hamiltonian is c harac teriz ed by the c oupling strength, we c an also desc ribe the c oarse-graining as a transformation in the c oupling parameter. Let K be the c oupling strength of the original system and R be the relevant transformation. The new c oupling strength is K′ = RK. This c oarse-graining proc edure c ould be iterated, produc ing a sequenc e of c oupling parameters, eac h related to the previous one by the transformation R. The transformation defines a flow on parameter spac e. How does this help us asc ertain the c ritic al behavior of a system? If you look at an Ising system at its c ritic al point, you will see c lusters of c orrelated spins of all siz es. This is a manifestation of the diverging c orrelation length. Now squint, blurring out the smaller c lusters. The new blurry system that you see will have the same general struc ture as the old one. You will still see c lusters of all siz es. This sort of sc ale invarianc e is c harac teristic of c ritic al behavior. The system has no c harac teristic length sc ale. Coarse-graining produc es a new system that is statistic ally identic al to the old one. At this point, the Hamiltonian of the system remains the same under indefinite c oarse-graining, so it must be a fixed point in parameter spac e (i.e., a point Kf suc h that Kf = RKf ). The nontrivial (viz ., not K = 0 or K = ∞) fixed points of the flow c harac teriz e the Hamiltonian of the system at the c ritic al point, the point at whic h c orrelation length diverges and there is no c harac teristic sc ale for the system. The c ritic al exponents c an be c alc ulated by series expansions near the c ritic al point. Critic al exponents predic ted by renormaliz ation group methods agree with experiment muc h more than the predic tions of mean field theory. The same approac h c an be applied to systems with more c omplic ated Hamiltonians involving a number of different parameters. Some of these parameters will be relevant, whic h means they get bigger as the system is resc aled. If a system has a nonz ero value for some relevant parameter, then it will not settle at a nontrivial fixed point upon resc aling, sinc e resc aling will amplify the relevant parameter and therefore c hange the c ouplings in the system. At c ritic ality, then, the relevant parameters must be z ero. An example of a relevant parameter for the Ising system is Page 5 of 24

Turn and Face The Strange … Ch-Ch-Changes the reduc ed temperature t. If t = 0, the system c an flow to a nontrivial fixed point. However, if t is perturbed from z ero, the system will flow away from this c ritic al fixed point toward a trivial fixed point. So a c ontinuous transition only takes plac e when t = 0, whic h is at the c ritic al temperature. Other parameters might turn out to be irrelevant at large sc ales. They will get smaller and smaller with suc c essive c oarse-grainings, effec tively disappearing at mac rosc opic sc ales. This elimination of mic rosc opic degrees of freedom means that the renormaliz ation group transformation c an be irreversible (whic h would, stric tly speaking, make it a semi-group rather than a group), and there c an be attrac tors in parameter spac e. These are fixed points into whic h a number of mic rosc opic ally distinc t systems flow. This is the basis of universality, the shared c ritic al behavior of quite different sorts of systems. If the systems share a fixed point their c ritic al exponents will be the same, even if their mic rosc opic Hamiltonians are distinc t. The differenc es in the Hamiltonians are in irrelevant degrees of freedom that do not affec t the mac rosc opic critical behavior of the system. Systems that flow to the same nontrivial fixed point are said to belong to the same universality c lass. The liquid– gas transition in water is in the same universality c lass as the paramagnetism– ferromagnetism transition. They have the same c ritic al exponents, despite the evident differenc es between the systems. The differenc e between relevant and irrelevant parameters c an be c onc eptualiz ed geometric ally. In parameter spac e, if we restric t ourselves to the hypersurfac e on whic h all relevant parameters are z ero, so that the differenc es between systems on this hypersurfac e are purely due to irrelevant parameters, then all points on the hypersurfac e will flow to a single fixed point. Perturb the system so that it is even slightly off the hypersurfac e, however, and the flow will take it to a different fixed point. It is signific ant that the fixed point only appears when the system has no c harac teristic length sc ale. This is why the infinite partic le limit is c ruc ial to the renor-maliz ation group approac h. If the number of partic les is finite, then there will be a c harac teristic length sc ale set by the siz e of the system. Coarse-graining beyond this length will no longer give us statistic ally identic al systems. The possibility of invari-anc e under indefinite c oarse-graining requires an infinite system. The requirement for the thermodynamic limit in renormaliz ation group theory c an be perspic uously c onnec ted to the motivation for this limit in the standard statistic al mec hanic al story. The c orrelation length of a system near its c ritic al point c an be c harac teriz ed in terms of some sec ond derivative of a thermodynamic potential. For instanc e, in a magnetic system the range of c orrelations between parts of the system is proportional to the susc eptibility, a sec ond derivative of the free energy. On the thermodynamic treatment, the susc eptibility diverges as we approac h the c ritic al point, and ac c ording to the statistic al mec hanic al treatment this is impossible unless we are in the thermo-dynamic limit. This means the c orrelation length c annot diverge, as is required for renormaliz ation group methods to work, unless the system is infinite. 2. The Emergence o f Phase Transitio ns? All of the above should sound a little troubling. After all, the systems we are interested in, the systems in whic h we see phase transitions every day, are not infinite systems. Yet the physic s of phase transitions seems to make c ruc ial appeal to the infinitude of the systems modeled. It appears that, ac c ording to both statistic al mec hanic s and renormaliz ation group theory, phase transitions c annot oc c ur in finite systems. Additionally, the explanation of the universal behavior of systems near their c ritic al point seems to require the infinite idealiz ation. Considerations of this sort have led many authors to say that phase transitions are genuinely emergent phenomena, suggesting that statistic al mec hanic s c annot provide a full reduc tive ac c ount of phase transitions in finite systems. The eminent statistic al mec hanic Lebowitz says phase transitions are “paradigms of emergent behavior” (Lebowitz , 1999, S346) and the philosopher Liu says they are “truly emergent properties” (Liu, 1999, S92). Needless to say, if this c laim is c orrec t, phase transitions present a c hallenge to philosophers with a reduc tionist bent. The extent of this c hallenge depends on how we interpret the c laim of emergenc e. The c onc ept of “emergenc e” is notoriously slippery, interpreted differently by different authors. We will c onsider a number of different arguments for phase transitions being emergent, c orresponding to varying c onc eptions of emergenc e. What these arguments have in c ommon is that they all involve a rejec tion of what Andrew Melnyk has c alled “reduc tionism in the c ore sense” (Melnyk, 2003, 83). This is the intuitive c onc eption of reduc tion that underlies various more prec ise philosophic al ac c ounts of reduc tion. A theory Th reduc es to a lower-level theory Tl if all the nomic c laims made by Th c an be explained using only the resourc es of Tl and nec essary truths. Page 6 of 24

Turn and Face The Strange … Ch-Ch-Changes This c onc eption is deliberately vague, allowing for various prec isific ations depending on one's theory of explanation and how one delineates the explanatory resourc es available to a partic ular theory. One possible prec isific ation is Ernest Nagel's ac c ount of reduc tion (Nagel, 1979), whic h says that Tl reduc es Th if and only if the laws of the latter c an be deduc ed from the laws of the former in c onjunc tion with appropriate bridge laws. In this ac c ount the c ore sense of reduc tion has been filled out with a logic al empiric ist theory of explanation ac c ording to whic h the explanatory resourc es of a theory are the deduc tive c onsequenc es of its lawlike statements. It is important to rec ogniz e that reduc tionists are c ommitted to this ac c ount of reduc tion only insofar as endorse suc h a theory of explanation. The proper motivation for Nagel's theory lies in the extent to whic h it suc c essfully c aptures the c ore sense of reduc tion. In this c hapter we do not endorse any partic ular ac c ount of reduc tion. Instead we c onsider three broad ways in whic h the explanatory c onnec tion between a higher-level theory and a lower-level theory may break down, and examine the extent to whic h these explanatory breakdowns are manifested in the c ase of phase transitions. Whether we have a genuine explanatory failure in a partic ular instanc e will depend on the details of our ac c ount of explanation. Often, the reduc tionist may be able to avoid a c ounterexample by simply rec onc eiving what c ounts as an adequate explanation.1 However, c ertain instanc es will be regarded as explanatory failures under a wide variety of plausible ac c ounts of explanation, perhaps even under all plausible ac c ounts of explanation. The weaker the assumptions about explanation required for the c ounterexample to work, the stronger the c ase for emergentism. We c an arrange our examples of purported explanatory failure into a hierarc hy based on the c onstraints plac ed on an ac c ount of explanation. At the bottom of this hierarc hy (at least for the purposes of this c hapter) is conceptual novelty. This is the sort of “irreduc ibility” involved when there is some natural kind in the higher-level theory that c annot be equated to a single natural kind in the lower-level theory. It may be the c ase that the phenomena that c onstitute the higher-level kind can be individually explained by the lower-level theory, but the theory does not unite them as a single kind. Conc eptual novelty involves a failure of type– type reduc tion, but need not involve a failure of token– token reduc tion. In the c ase of phase transitions, it has been suggested that although one c an provide a perfec tly adequate explanation of individual transitions using statistic al mec hanic s, the theory does not distinguish these phenomena as a separate kind. For instanc e, from the perspec tive of statistic al mec hanic s, the transition from ic e to water in a finite system as we c ross 273.16 K is not qualitatively different from the transition from c old ic e to slightly warmer ic e as we c ross 260 K, at least if something like the standard story is c orrec t. The only differenc e is that the thermodynamic potentials c hange a lot more rapidly in the former situation than in the latter, but they are still analytic , so this is merely a differenc e of degree, not a differenc e of kind. There are two tac ks one c an take in response to this observation. The first is that this is a c ase where statistic al mec hanic s c orrec ts thermodynamic s. Just as it showed us that the sec ond law is not in princ iple exc eptionless, it shows us that rigorous separation of phases, the only phenomenon worthy of the name “phase transition,” is only possible in infinite systems. This view of the emergenc e of phase transitions is expressed by Kadanoff when he says that “in some sense phase transitions are not exac tly embedded in the finite world but, rather, are produc ts of the human imagination” (Kadanoff 2009, 778). Thermodynamic s c lassifies a set of empiric al phenomena as phase transitions, involving a qualitatively distinc t type of c hange in the system. Statistic al mec hanic s reveals that these phenomena have been misc lassified. They are not genuinely qualitatively distinc t and should not be treated as a separate natural kind. This response does not appear to pose muc h of a threat to reduc tionism. It may be true that thermodynamic s has not been reduc ed to statistic al mec hanic s in a stric t Nagelian sense, but this seems like muc h too restric tive a c onc eption of reduc tion. There are many paradigmatic c ases of sc ientific reduc tion where the reduc ing theory explains a c orrec ted version of the reduc ed theory, not the theory in its original form. This c orrec tion may often involve dissolving inappropriate distinc tions. If this is all there is to the c hallenge of c onc eptual novelty, it is not muc h of a c hallenge. However, one might want to resist this eliminativism and rejec t the notion that thermodynamic s has misc lassified phenomena. Perhaps the right thing to say is that at the thermodynamic level of desc ription it is indeed appropriate to have a distinc t kind c orresponding to phase transitions in finite systems. But the appropriateness of this kind is invisible at the statistic al mec hanic al level of desc ription, sinc e statistic al mec hanic s does not have the resourc es to c onstruc t suc h a c lass. This is a more substantive c hallenge to reduc tionism, akin to c ases of multiple realiz ability. As an analogy, c onsider that from the perspec tive of our molec ular theory there is no natural kind (or indeed finite disjunc tion of kinds) c orresponding to the c ategory “c an opener.” It seems implausible that we will be Page 7 of 24

Turn and Face The Strange … Ch-Ch-Changes able to delineate the c lass of c an openers using only the resourc es of our mic rosc opic theory. Yet we do not take this to mean that our mic rosc opic theory c orrec ts our mac rosc opic theory, demonstrating that c an openers do not exist as a separate kind. Can openers do exist. They are an appropriate theoretic al kind at a c ertain level of desc ription. Similarly, the fac t that statistic al mec hanic s does not have the resourc es to delineate the c lass of finite partic le phase transitions need not lead us to c onc lude that this c lassific ation is bogus. How might the reduc tionist respond to c onc eptual novelty of this sort? One response would be to develop a sense of explanation that makes reduc tion c ompatible with multiple realiz ation. Even though statistic al mec hanic s does not group phase transitions together the way that thermodynamic s does, it is still able to fully explain what goes on in individual instanc es of phase transition. Perhaps the existenc e of individual explanations in every c ase c onstitutes an adequate explanation of the nomic pattern desc ribed by thermodynamic s. If this is the c ase, the c ore sense of reduc tion is satisfied. One does not need to look at phase transitions to notic e that any c laim about the reduc tion of thermodynamic s to statistic al mec hanic s must be based on a c onc eption of reduc tion that is c ompatible with multiple realiz ability. Temperature, that most basic of thermodynamic properties, is not (the c laims of numerous philosophers notwithstanding) simply “mean molec ular kinetic energy.” It is a multiply realiz able func tional kind. If our notion of reduc tion prec ludes the existenc e of suc h properties, then the projec t of reduc ing thermodynamic s c annot even get off the ground. To us, this seems like the c orrec t response to c laims of emergenc e based on the c onc eptual novelty of phase transitions. If this is all it takes for emergenc e, then prac tic ally every thermodynamic property is emergent. Perhaps the emergentist is willing to bite this bullet, but we think it is more plausible that the argument from c onc eptual novelty to emergenc e relies on a muc h too restric tive c onc eption of sc ientific explanation. It is, however, worth noting another line of response. It may be the c ase that a c lass of finite partic le phase transitions c an be c onstruc ted within statistic al mec hanic s that overlaps somewhat (but not c ompletely) with the ther-modynamic c lassific ation. This would be a c ase of statistic al mec hanic s c orrec ting thermodynamic s, but not by eliminating the phenomenon of phase transitions in finite systems. Instead, statistic al mec hanic s would redefine phase transitions in a manner that preserves our judgments about a number of empiric al instanc es of the phenomenon. If suc h a redefinition c ould be engineered, phase transitions would not be c onc eptually novel to thermodynamic s. The prospec ts for this strategy are disc ussed in sec tion 3.1. Let us suppose our c onc eption of reduc tion is broad enough that mere c onc eptual novelty does not indic ate a failure of reduc tion. We ac c ept with equanimity that under c ertain c onditions it might be appropriate to model phenomena using a c onc eptual voc abulary distinc t from that of our reduc ing theory. For instanc e, at a suffic iently c oarse-grained level of desc ription a c ertain set of thermody-namic transformations is fruitfully modeled as exhibiting singular behavior, and appropriately grouped together into a separate natural kind. However, one might not think that a fully reduc tive explanation has been given unless one c an explain using the resourc es of the reduc ing theory why this model is so effec tive under those c onditions. Why does modeling a finite partic le phase transition as nonanalytic work so well at the thermodynamic level of desc ription if finite systems c annot exhibit non- analytic ities at the statistic al mec hanic al level of desc ription? If we c annot give suc h an explanation, we have another potential variety of emergenc e: explanatory irreducibility. To give an idea of the kind of story we are looking for, c onsider the infinite idealiz ation involved in explaining the extensivity of c ertain thermodynamic properties. Many thermodynamic properties are extensive, suc h as the entropy, internal energy, volume, and free energy. What this means is that if we divide a system into mac rosc opic parts, the values of those properties behave in an additive way. Loosely put, if we double the siz e of the system (that is, double internal energy, partic le number, volume), then we double that system's extensive properties (e.g., the entropy).2 Intensive properties, by c ontrast, do not sc ale this way; for example, if we double the siz e of a system, we do not double the pressure. Extensivity and intensivity are features usefully employed by phenomenologic al thermodynamic s. However, when we look at a system mic rosc opic ally, we quic kly see that no finite system is ever stric tly extensive or intensive. Correlations exist between the partic les in one part of a system and another part. If we want to reproduc e the thermody-namic distinc tion exac tly, we are stymied: no matter how large the system, if it is finite, surfac e effec ts c ontribute to the partition func tion, whic h will mean that systems’ energies and entropies c annot be neatly halved. For instanc e, if we define the entropy as a func tion over the joint probability distributions involved (as with the Gibbs entropy), we see that the entropy is extensive only when the two subsystems are probabilistic ally independent of one another. The only plac e we c an reproduc e the sharp distinc tion is by going to the thermodynamic limit. There we c an define a variable f as extensive if f goes to infinity 3 Page 8 of 24

Turn and Face The Strange … Ch-Ch-Changes as we approac h the thermodynamic limit while f/V is c onstant in the limit, where V is the volume of the system.3 Strictly speaking, only in infinite systems are entropy, energy, and so on, truly extensive. Does this fact imply that there is a great mystery about extensivity, that exten-sivity is truly emergent, that thermodynamic s does not reduc e to finite N statistic al mec hanic s? We suggest that on any reasonably unc ontentious way of defining these terms, the answer is no. We know exac tly what is happening here. Just as the sec ond law of thermodynamic s is no longer stric t when we go to the mic rolevel, neither is the c onc ept of extensivity. The notion of extensivity is an idealiz ation, but it is one approximated well by finite partic le statistic al mec hanic s. For boxes of length l c ontaining partic les interac ting via short-range forc es, the surfac e effec ts sc ale as l2 and the volume as l3 . Surfac e effec ts bec ome less and less important as the system gets larger. Beings restric ted to mac rosc opic physic s would do well to c all upon the extensive/intensive distinc tion, sinc e in most c ases the impac t of surfac e effec ts would be well beyond the prec ision of measurements made by suc h beings. Here we see that extensivity in finite systems is c onc eptually novel to thermodynamic s. It does not exist in statistic al mec hanic s. However, leaving the story there is unsatisfac tory. We need a further ac c ount, from a statistic al mec hanic al perspec tive, of why this new c onc ept works so well in thermodynamic s. And indeed suc h a story is forthc oming. It relies c ruc ially on the fac t that the resolution of our measurements is limited, but this in itself does not, or at least should not, derail the reduc tionist projec t. As long as we have a story that explains why beings with suc h limitations c ould fruitfully desc ribe suffic iently large systems as extensive—a story in terms of the c omponents of the system and their organiz ation, and how relevant quantities sc ale as the system gets larger—we do not have a genuine c hallenge to reduc tionism in the c ore sense. The question is whether a similar sort of explanation is available to ac c ount for the effic ac y of the infinite idealiz ation involved in the statistic al mec hanic al analysis of phase transitions. If there is not, we would have a c ase for emergenc e. There would be something about the systems under c onsideration that c ould not be ac c ounted for reduc tively, namely, the fac t that their behavior at a phase transition c an, under c ertain c onditions, be adequately modeled as the behavior of an infinite system. This feature of finite systems is c ruc ial to understanding their behavior at phase transitions, so if it c annot be explained it would be legitimate to say that phase transitions are emergent. In sec tion 3.2 we examine the possibility of a reduc tive explanation of the effic ac y of the infinite idealiz ation. Modeling the behavior of partic ular systems is not the only func tion of the infinite idealiz ation in the study of phase transitions. The idealiz ation plays a c entral role in the renormaliz ation group explanation for universal behavior at the c ritic al point. As we have disc ussed above, universal behavior is ac c ounted for by the presenc e of stable fixed points in the spac e of Hamiltonians, eac h of whic h is the terminus of a number of different renormaliz ation flow trajec tories. This sort of explanation raises spec ial problems that do not arise when we c onsider the sort of infinite idealiz ation involved in the assumption of extensivity. There we have a property that, as it turns out, c an only be approximated by finite systems. It is only ac tually instantiated in infinite systems. However, the property itself c an be c harac teriz ed without rec ourse to the infinite idealiz ation. We c ould in princ iple c onstruc t an explanation of why a finite thermodynamic system approximates extensive behavior without any appeal to the infinite idealiz ation. The idealiz ation gives us a model of a genuinely extensive system, but it is not essential to an understanding of why it is useful to treat mac rosc opic finite systems as extensive. It appears that the situation is different when we c onsider the renormaliz ation group explanation of universality. There, the infinite idealiz ation plays a different role. Talking about how a partic ular large finite system approximates the behavior of an infinite system will not be helpful, bec ause universality is not about the behavior of individual systems, finite or infinite. It is a c harac teristic of c lasses of systems. The renormaliz ation group method explains why physic al systems separate into distinc t universality c lasses, and it explains this in terms of c ertain struc tural features of the spac e of systems, the fixed points of the renormaliz ation flow. It is the existenc e of these features, and their c onnec tion to the phenomenon of universality, that requires the infinite idealiz ation. We might be able to give an ac c ount of why a partic ular large finite system approac hes very c lose to a fixed point as it is resc aled, approximating the behavior of an infinite system, but this will not tell us why this behavior matters. In order to see the c onnec tion between approac hing a fixed point and exhibiting universal behavior, we need the infinite idealiz ation. This argument is made in Batterman (2011). We address it in sec tion 4. In a c ase of explanatory irreduc ibility the higher-level theory models a partic ular phenomenon in a c onc eptually novel manner, and the effic ac y of this model c annot be explained by the lower-level theory. However, this does Page 9 of 24