Failure of thermodynamics near a phase transition

Abstract

We investigate the relation between various statistical ensembles of finite systems. If ensembles differ at the level of fluctuations of the order parameter, we show that the equations of states can present major differences. A sufficient condition for this inequivalence to survive at the thermodynamical limit is worked out. If energy consists in a kinetic and a potential part, the microcanonical ensemble does not converge towards the canonical ensemble when the partial heat capacities per particle fulfill the relation c −1PACS numbers: 05.20. Gg, 64.10.+h, 05.70.Fh In most textbooks the equivalence between the different statistical ensembles is demonstrated at the thermodynamical limit through the Van Hove theorem [1]. Indeed ensembles differ at the level of fluctuations which are generally believed to induce small corrections in finite systems and to become negligeable at the limit of infinite systems.In this paper we will show that this might not be always the case. For finite systems, two ensembles which put different constraints on the fluctuations of the order parameter lead to qualitatively different equations of states close to a first order phase transition. As an example the microcanonical heat capacity may diverge to become negative while the canonical one remains always positive and finite [2,3]. Such inequivalences may survive at the thermodynamical limit for systems involving long range forces [4,5]. Looking at the general properties of the order parameter distribution a sufficient condition for this behavior to show up can be explitely worked out.Let us first concentrate on finite systems. For simplicity we will consider the microcanonical and the canonical ensemble characterized by the energy E and the temperature β −1 respectively, but our discussion is valid for any couple of conjugated extensive and intensive variables.The microcanonical ensemble is characterized by the level density W (E) and the entropy S = log W .The caloric curve is then T −1 = ∂ E S. The canonical partition sum is the Laplace transform of W : Z β = dEW exp(−βE). In this article we will assume that the partition sum converges; this is not always the case [6] and indeed the impossibility to normalize the distribution W exp(−βE) is already a known case of ensemble inequivalence.In finite systems, the canonical ensemble differs from the microcanonical one since it does not correspond to a unique energy but to a distribution P β (E) = exp(S(E) − βE − log Z β ). If P β has a single maximum the average energy E β = −∂ β ln Z β can also be computed using a saddle point approximation around the most probable energy E βwith g β (x) = c 0 +c 3 x 3 +c 4 x 4 +. . . . If P β is symmetric,The definition of saddle impliesmeaning that the microcanonical caloric curve T (E) exactly coincides with the canonical one β −1 ( E ). However in a finite system the distribution may be not symmetric so that the two curves can be shifted :.. withg β the series of the odd terms of g β . However, the shift δ is in most cases small so that when P β h...

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