This paper reviews a number of fundamental connections that exist between
nonequivalent microcanonical and canonical ensembles, the appearance of
first-order phase transitions in the canonical ensemble, and thermodynamic
metastable behavior.Comment: 4 pages, RevTeX, 1 figure. Contribution to the Proceedings of the
  31st Workshop of the International School of Solid State Physics
  ``Complexity, Metastability and Nonextensivity'', held at the Ettore Majorana
  Foundation and Centre for Scientific Culture, Erice, Sicily, Italy, July
  2004. Edited by C. Tsallis, A. Rapisarda and C. Beck. To be published by
  World Scientific, 200

Ellis, Richard S.

Touchette, Hugo

English

arXiv

This paper reviews a number of fundamental connections that exist between nonequivalent microcanonical and canonical ensembles, the appearance of first-order phase transitions in the canonical ensemble, and thermodynamic metastable behavior

TOUCHETT, H

Ellis, RS

ScholarWorks@UMass Amherst

University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Publication Series Mathematics and Statistics 2005 Nonequivalent ensembles and metastability H TOUCHETT RS Ellis University of Massachusetts - Amherst, rsellis@math.umass.edu Follow this and additional works at: https://scholarworks.umass.edu/math_faculty_pubs  Part of the Physical Sciences and Mathematics Commons Recommended Citation TOUCHETT, H and Ellis, RS, "Nonequivalent ensembles and metastability" (2005). Complexity, Metastability and Nonextensivity. 1192. https://doi.org/10.48550/arXiv.cond-mat/0501390 This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact scholarworks@library.umass.edu. arXiv:cond-mat/0501390v1  [cond-mat.stat-mech]  17 Jan 2005Nonequivalent ensembles and metastability∗Hugo Touchette1, † and Richard S. Ellis2, ‡1School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK2Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA(Dated: November 13, 2018)This paper reviews a number of fundamental connections that exist between nonequivalent micro-canonical and canonical ensembles, the appearance of first-order phase transitions in the canonicalensemble, and thermodynamic metastable behavior.I. INTRODUCTIONThe goal of this short paper is to trace a line of relation-ships that goes from the phenomenon of nonequivalentmicrocanonical and canonical ensembles to that of ther-modynamic metastability. Our approach will aim for themost part at stressing the physics of these relationships,but care will also be taken to formulate them in a precisemathematical language. However, due to the limitationin available space, many mathematical details will haveto be left aside, including the proofs of all the resultsstated here. References containing these proofs, whenthey exist, will be mentioned to assist the reader. An-other more complete paper that treats these relationshipswith full mathematical details is also in preparation,12based on the recent doctoral dissertation of one of us.21Far from being exhaustive, we hope that this short re-view can serve as a starting point in the literature for thereader interested in knowing about nonequivalent ensem-bles, as well as those interested in phase transitions andmetastable behavior in many-body systems. These pro-ceedings are a testament to the fact that there remain atpresent many unsolved problems related to metastability,and it is our belief that what has been learned in stud-ies of nonequivalent ensembles could yield useful cluesfor solving these problems. Perhaps the most obviousof these clues is the fact that many of the systems de-scribed in these pages (see, e.g., the contributions on theHMF model) exhibit negative values of the heat capac-ity at fixed energies at the same time that they exhibitmetastable states. The negativity of the heat capacity atfixed energy is well known to be related to the nonequiva-lence of the microcanonical and canonical ensembles. It isalso, as we will see here, a direct indication of metastablebehavior.II. NONEQUIVALENT ENSEMBLESThe equivalence of the microcanonical and canonicalensembles is most usually explained by saying that al-though the canonical ensemble is not a fixed-mean-energyensemble like the microcanonical ensemble, it must ‘con-verge’ to a fixed-mean-energy ensemble in the thermody-namic limit, and so must become or must realize a micro-canonical ensemble in that limit.11,22 This explanation isnot far from being entirely valid, but there is a problemwith it: the canonical ensemble may not in fact realizeat equilibrium all the mean energies that can be real-ized in the microcanonical ensemble.22 In other words,the range of the equilibrium mean energy uβ realized inthe canonical ensemble by fixing the inverse temperatureβ may be only a subset of the range of definition of themean energy u itself. If this is the case, then the micro-canonical ensemble must be richer than the canonical en-semble because there are values of the mean energy thatcan assessed within the microcanonical ensemble, but notwithin the canonical ensemble. The two ensembles musttherefore be nonequivalent.To see how this possibility can arise, and how it isrelated in fact to the nonconcavity of the microcanonicalentropy function, let us introduce some notation. Weconsider, as is usual in statistical mechanics, an n-bodysystem with Hamiltonian U and mean entropy s(u) =S(U)/n, where u = U/n is the mean energy. To stateour first result, we need to define an important conceptin convex analysis known as a supporting line.9,19 This isdone as follows: we say that s admits a supporting line atu if there exists β ∈ R such that s(v) ≤ s(u) + β(v − u)for all admissible u. From a geometric point of view,the requirement of a supporting line should be clear: itmeans that we can draw a line above the graph of s(u)that passes only through the point (u, s(u)); see Fig. 1.The slope of this line is β.Theorem 1. Let uβ be the value of the mean Hamil-tonian realized at equilibrium in the canonical ensemblewith inverse temperature β. (There can be more thanone equilibrium value.) Then, for any admissible meanenergy value u, there exists β such that uβ = u if andonly if s admits a supporting line at u with slope β.This simple result seems to have floated in the mindsof physicists for a long time. It is implicit, for exam-ple, when considering the physical meaning of first-orderphase transitions in the canonical ensemble and theirconnection with nonconcave entropies.6,13,14,15,16,17,18,20However, to the best of our knowledge, there has neverbeen a clear formulation of this result until recently.21,22This can be explained in part by the fact that the conceptof a supporting line is not well known in physics.The full application of our first theorem is presentedin Fig. 1 which shows the plot of a generic entropy func-tion having a nonconcave part. This figure depicts threepossible cases:Typeset by REVTEX2us(u)(a) (b) (c)ul uh¯cFIG. 1: (a) Concave point of the microcanonical entropy function s(u) which admits a supporting line. (b) Two concave pointsof s(u) which admit the same supporting line. (c) Nonconcave point of s(u) which does not admit a supporting line.(a) Mean energy value u for which s admits a sup-porting line. In this case, the value u can be realizedat equilibrium in the canonical ensemble by setting βequal to the slope of the supporting line passing through(u, s(u)). We naturally expect in this case to see themicrocanonical ensemble at u give the same equilibriumpredictions as the canonical ensemble at β since the lat-ter ensemble reduces to a single-mean-energy ensemblewith uβ = u. Note that β must be such that s′(u) = β ifs is differentiable.(b) There exists a single supporting line that touchestwo points of the graph of the microcanonical entropy. Inthis case, not one but two values of the mean energy—e.g., ul and uh in Fig. 1—are realized at equilibrium inthe canonical ensemble for β corresponding to the slopeof the supporting line. This situation, as will be clearin the next section, corresponds to a state of coexistingphases which universally signals the onset of a first-orderphase transition in the canonical ensemble.(c) Mean energy u for which s admits no supportingline. This also applies for all u ∈ (ul, uh). Theorem 1states for this case that the canonical ensemble must beblind to the properties of the microcanonical ensemblesince it cannot realize at equilibrium any of the meanenergies u ∈ (ul, uh) for any values of β. This meansin particular that the standard thermodynamic relationβ = s′(u) ceases to be valid in this region.11,22 The nexttheorem relates this case of nonequivalent ensembles withthe occurrence of negative values of the heat capacity inthe microcanonical ensemble.14,15Theorem 2. Define the microcanonical heat capacity atthe mean energy value u by c(u) = −s′(u)2s′′(u)−1. Ifc(u) < 0, then s does not have a supporting line at u.This result is a new formulation—again because theuse of supporting lines—of an old result that relates thenegativity of c with the nonequivalence of the micro-canonical and canonical ensembles. Usually what is con-cluded is that these two ensembles must be nonequiva-lent when c < 0 because the heat capacity can never benegative in the canonical ensemble.14,15,17,18,20 Our for-mulation has the advantage of stressing the physical rootof negative heat capacities, namely that the mean ener-gies u which are such that c(u) < 0 are not equilibriummean energies in the canonical ensemble. This point willbe discussed further in Section IV. For now, let us notein closing this section that the negativity of c is only asufficient condition for ensemble nonequivalence, not anecessary one.11,22 Thus it is not true that the canonicalensemble is blind to the microcanonical ensemble onlyfor those mean energy values u such that c(u) < 0, as isoften claimed.14,15,17,18,20 As we have seen, the canonicalensemble is in fact blind to the microcanonical ensem-ble for all u at which s admits no supporting lines, andthat, in general, comprises more values u than only thosehaving c(u) < 0; see, e.g., Fig. 1.III. NONEQUIVALENT ENSEMBLES ANDFIRST-ORDER PHASE TRANSITIONSThe previous section makes it clear that what is re-sponsible for the nonequivalence of the microcanonicaland canonical ensembles is the occurrence of a first-orderphase transition in the canonical ensemble. To be sure,just replace the word ‘blind’ with the word ‘skip’ to ob-tain a sentence such as: the microcanonical and canonicalensembles are nonequivalent because the canonical en-semble skips over an interval of mean energies which canbe accessed microcanonically.14,15,18,20 The inverse tem-perature at which the canonical ensemble skips over themicrocanonical ensemble corresponds, not surprisingly,to the inverse temperature at which a first-order phasetransition appears. This is the subject of the next theo-rem which relates the nonconcavity property of s(u) withthe differentiability property of the free energy functionϕ(β), the central thermodynamic quantity of the canon-ical ensemble which is taken here to be defined by thelimitϕ(β) = limn→∞−1nlnZn(β), (1)where is Zn(β) is the standard n-body partitionfunction.9,11Theorem 3. Assume that s admits no supporting linesfor all u ∈ (ul, uh). Then ϕ is non-differentiable at a3critical value βc equal to the slope of the supporting linethat bridges ul and uh. The left- and right-derivatives ofϕ at βc equal uh and ul, respectively.This theorem is a direct result of the fact thatϕ(β) is the Legendre-Fenchel transform of s(u), whethers(u) is concave or not, and some basic propertiesof these transforms.9,11,13 It can be found in manyworks14,15,17,18,20 which do not define, however, the con-cavity of s(u) in terms of supporting lines. This is a mi-nor omission because most of these works use an equiv-alent method for defining the range of nonconcavity ofs(u) based on the so-called Maxwell’s construction.14,15In any case, it is clear in all the works just cited thatthe nonequivalence of the microcanonical and canonicalensembles arises as a consequence of first-order phasetransitions in the canonical ensemble. The nonconcav-ity of s(u), which translates into a ‘back-bending’ shapeof s′(u), is in fact sometimes taken as a definition or aprobe of canonical first-order phase transitions.6,14,15,16The opposite is also possible; that is, it is possible to re-late the absence of a first-order phase transition in thecanonical ensemble with the equivalence of the micro-canonical and canonical ensembles.11 This is done in thenext theorem.Theorem 4. If ϕ is differentiable at β, then s admits astrict supporting line that touches the graph of s only atu = ϕ′(β).This result implies the following standard result: if ϕis differentiable at β, then u = ϕ′(β) is the unique meanenergy value realized at equilibrium in the canonical en-semble with inverse temperature β.IV. NONEQUIVALENT ENSEMBLES ANDMETASTABILITYThe last set of results that we will discuss directly per-tains to the mean energies which can be assessed mi-crocanonically but not canonically. What we want toshow is that these nonequivalent mean energies corre-spond to nonequilibrium critical mean energies of thecanonical ensemble. This is somewhat obvious giventhat they cannot be equilibrium mean energies; however,what we want to discuss more specifically is the phys-ical nature of these nonequilibrium critical points. Todo so, we have to note that the values uβ of the meanenergy that are realized at equilibrium in the canonicalensemble at β are, by definition, the global minimum ofthe function Fβ(u) = βu − s(u),6,16 which we call thenonequilibrium free energy function.21,22 This implies inparticular that uβ must satisfy ∂uFβ(uβ) = 0, or equiva-lently β = s′(uβ), assuming that s is differentiable. Notehowever—and this is the crucial point here—that not allthe points u satisfying β = s′(u) may globally minimizeFβ(u); some of these critical points may actually corre-spond to local minimum of Fβ(u) or even local maxi-mum of Fβ(u). To determine the precise nature of thesenonequilibrium canonical critical points, we can look atthe sign of the second u-derivative of Fβ(u) to obtain thefollowing result.Theorem 5. Suppose that s does not admit a supportingline at u.(a) If c(u) > 0, then u is a metastable mean energy ofthe canonical ensemble, in the sense that it is a local butnot global minimum of Fβ(u) for β = s′(u).(b) If c(u) < 0, then u is an unstable mean energyof the canonical ensemble, in the sense that it is a localmaximum of Fβ(u) for β = s′(u).While this result applies to the mean energy, it is in-teresting to see if anything can be said about generalmacrostates: e.g., the magnetization or the distributionof states. We all know, for instance, that phase tran-sitions in spin systems can be revealed at the level ofthe mean energy (thermodynamic level) or at the levelof the magnetization (macrostate level) since both levelsare related in a one-to-one fashion. Is the same true formetastability? That is, can the metastable behavior of asystem be revealed at the macrostate level? If so, can thismacrostate level of metastability be related to the ther-modynamic level of metastability defined with respect tothe mean energy?The answer to these questions is yes, so long as we areconcerned with mean-field systems, which are basicallysystems for which the Hamiltonian U can be expressedas a function of some macrostate m of interest.5,21 Inthis case, we can formulate the following result about themacrostate values mu which are realized at equilibriumin the microcanonical ensemble with mean energy u, butnot in the canonical ensemble for any β. The result isformulated in terms of the nonequilibrium free energyFβ(m) which is the macrostate generalization of Fβ(u).21Theorem 6. Suppose that s does not admit a supportingline at u.(a) If c(u) > 0, then mu is a metastable macrostate ofthe canonical ensemble, in the sense that it is a local butnot global minimum of Fβ(m) for β = s′(u).(b) If c(u) < 0, then mu is an unstable macrostate ofthe canonical ensemble, in the sense that it is a saddle-point of Fβ(m) for β = s′(u).What this results says physically is that a macrostatevalue mu which is stable in the microcanonical ensemblecan become unstable and thus decay in time if we releasethe energy constraint and fix the inverse temperature in-stead, as in the canonical ensemble. The precise way inwhich mu decays in the canonical ensemble to a differentequilibrium value mβ is determined by the local geom-etry of Fβ(m) around mu which is determined, in turn,by the sign of c(u). For more details on this result, thereader is referred to two papers1,11 which contain the re-sult of Theorem 6 in a more or less conjectured form. Aproof of this theorem can be found in a recent proceed-ings paper of Campa and Giansanti.5 Another proof willbe presented elsewhere.124V. CONCLUDING REMARKSThe present paper hardly exhausts the subject ofnonequivalent ensembles and metastability. In going fur-ther, we could have reviewed recent works on the dynam-ics of nonequivalent states in the canonical ensemble,2,4,18as well as the dynamical stability of these states10 whichis discussed, for example, in Anteneodo’s contribution tothese proceedings using an approach based on Vlasov’sequation. We could have alluded also to the fact thatnonconcave entropies are seen in fields as disconnectedas string theory7 and multifractal analysis.3 Finally, wecould have mentioned our recent work on a generaliza-tion of the canonical ensemble which aims at convertingunstable and metastable states of the canonical ensembleinto stable, equilibrium states of a modified canonical en-semble so as to recover equivalence with the microcanon-ical ensemble.8 Research is ongoing on this topic.AcknowledgmentsOne of us (H.T.) would like to thank the organizingcommittee of the Complexity, Metastability and Nonex-tensivity Conference for its hospitality and for financialsupport. The research of H.T. was supported by NSERC(Canada) and the Royal Society of London, while thatof R.S.E. was supported by the National Science Foun-dation (NSF-DMS-0202309).∗ Contribution to the Proceedings of the 31st Workshop of the International School of Solid State Physics “Complexity,Metastability and Nonextensivity”, held at the Ettore Majorana Foundation and Centre for Scientific Culture, Erice, Sicily,Italy, July 2004. Edited by C. Tsallis, A. Rapisarda and C. Beck. To be published by World Scientific, 2005.† Electronic address: htouchet@alum.mit.edu‡ Electronic address: rsellis@math.umass.edu1 M. Antoni, S. Ruffo, A. Torcini, Phys. Rev. E 66, 025103, (2002).2 M. Antoni, S. Ruffo, A. Torcini, Europhys. Lett. 66, 645 (2004).3 C. Beck, F. Schlögl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1993).4 F. Bouchet, Phys. Rev. E 70, 036113 (2004).5 A. Campa, A. Giansanti, Physica A 340 , 170 (2004).6 Ph. Chomaz, F. Gulminelli, V. Duflot, Phys. Rev. E 64, 046114 (2001).7 M.A. Cobas, M.A.R. Osorio, M. Suárez, Phys. Lett. B 601, 99 (2004).8 M. Costeniuc, R.S. Ellis, H. Touchette, B. Turkington, cond-mat/0408681.9 R.S. Ellis, K. Haven, B. Turkington, J. Stat. Phys. 101, 999 (2000).10 R.S. Ellis, K. Haven, B. Turkington, Nonlinearity 15, 239 (2002).11 R.S. Ellis, H. Touchette, B. Turkington, Physica A 335, 518 (2004).12 R.S. Ellis, H. Touchette, Nonequivalent ensembles, metastability, and first-order phase transitions, in preparation (2005).13 G.L. Eyink, H. Spohn, J. Stat. Phys. 70, 833 (1993).14 D.H.E. Gross, Phys. Rep. 279, 119 (1997).15 D.H.E. Gross, Microcanonical Thermodynamics: Phase Transitions in “Small” Systems, Lecture Notes in Phyics, Vol. 66(World Scientific, Singapore, 2001).16 F. Gulminelli, Ph. Chomaz, Phys. Rev. E 66, 046108 (2002).17 P. Hertel, W. Thirring, Ann. Phys. (NY) 63, 520 (1971).18 D. Lynden-Bell, Physica A 263, 293 (1999).19 R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970).20 W. Thirring, Z. Physik. 235, 339 (1970).21 H. Touchette, Equivalence and nonequivalence of the microcanonical and canonical ensembles: a large deviations study,Ph.D. Thesis, McGill University (2003).22 H. Touchette, R.S. Ellis, B. Turkington, Physica A 340, 138 (2004).

Nonequivalent ensembles and metastability

This paper reviews a number of fundamental connections that exist between nonequivalent microcanonical and canonical ensembles, the appearance of first-order phase transitions in the canonical ensemble, and thermodynamic metastable behavior.81-8

Nonequivalent ensembles and metastability

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