Statistical Mechanics of Equilibrium and Nonequilibrium Phase Transitions: The Yang–lee Formalism

Abstract

Showing that the location of the zeros of the partition function can be used to study phase transitions, Yang and Lee initiated an ambitious and very fruitful approach. We give an overview of the results obtained using this approach. After an elementary introduction to the Yang-Lee formalism, we summarize results concerning equilibrium phase transitions. We also describe recent attempts and breakthroughs in extending this theory to nonequilibrium phase transitions. YANG-LEE FORMALISM FOR PHASE TRANSITIONS 5particle separations, i.e., in terms of the interaction potential, |u(r)| C 1 /r d+ε as r → ∞ (with C 1 > 0 and ε positive constants), where d is the dimension of the system. Such systems are currently called short-range interaction systems. (ii) u(r) has to have a repulsive part increasing sufficiently rapid at small interparticle distances (preventing the system from collapse at high particle number densities). Throughout this paper, we shall consider systems with a hard-core interparticle repulsion at small distances, u(r) = ∞ for r < r 0 ∼ b 1/d (where b is the single-particle excluded volume), which means that a system of volume V can accommodate a finite maximum number of particles M = V /b. c (iii) Finally, the interaction potential has to be everywhere bounded from below, u(r) −u 0 whatever r (with u 0 a positive constant).Note a rather general result concerning the one-dimensional equilibrium systems with short range interactions, namely van Hove's theorem 1,5,7 , according to which no phase transition is possible in such systems, in contrast to what happens generically in corresponding nonequilibrium systems, see Ref. 8 for a brief review. A recent critical discussion 9 of this result allowed to highlight some exceptions to this theorem, but we shall not be concerned here with these rather pathological situations.Many physical situations can be modeled by interaction potentials with the characteristics (i) -(iii) above. There are, however, a few exceptions. The most salient example is that of the systems with long-range interactions (or non-additive systems), which include, e.g., systems with gravitational or Coulombian forces (see Ref. 6 for a modern review of their (thermo)dynamical properties studies). Such systems exhibit an unequivalence of the statistical ensembles, and it is not clear yet how to apply the concepts of the Yang-Lee theory (as described below) to their phase transitions. We will not address them further here. A limiting case of such systems is that of mean-field interactions, on which we will briefly comment in Secs. 7.2, 12, 13.3. Yang-Lee zeros of the grand-canonical partition function: the general frameworkWe shall address here two problems, namely:(A) The location of the phase transition point: How, when knowing the partition function (in the thermodynamic limit), can one locate a phase transition point by investigating the zeros of this partition function with respect to the control parameter of the system.(B) The characteristics of the transition: How can one extract informa...

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