The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

Abstract

We present a pedagogical account of the Lee-Yang theory of equilibrium phase transitions and review recent advances in applying this theory to nonequilibrium systems. Through both general considerations and explicit studies of specific models, we show that the Lee-Yang approach can be used to locate and classify phase transitions in nonequilibrium steady states. I IntroductionIn this work we seek a mathematical understanding of phase transitions in the steady state of stochastic many-body systems. Systems at equilibrium with their environment provide examples of such steady states, and the mechanisms underlying equilibrium phase transitions are long known and understood. Experimentally, one can distinguish between two types of transition: the first-order transition at which there is phase coexistence, e.g. between a high-density solid and a low-density fluid, and the continuous transition at which fluctuations and correlations grow to such an extent as to be macroscopically observable.From a thermodynamical perspective, one can understand first-order transitions by associating with each phase a free energy. For a given set of external parameters, the phase 'chosen' by the system is that with the lowest free energy, and so a phase transition occurs when the free energies of two (or more) phases are equal. The sudden changes in macroscopically measurable quantities that take place at first-order transitions are described mathematically as discontinuities in the first derivative of the free energy. Discontinuities in higher derivatives relate to continuous (higherorder) phase transitions.The tools of equilibrium statistical physics allow onein principle at least-to express the free energy solely in terms of microscopic interactions. More specifically, the free energy is given by the logarithm of a partition function, a quantity that normalises the steady-state probability distribution of microscopic configurations. Initially it was not universally accepted that this approach could faithfully describe phase transitions, in particular the first-order solidfluid transition [1]. In order to show that the statistical mechanical approach can reproduce the correct discontinuities in the free energy at a first-order transition, Lee and Yang [2] introduced a description of phase transitions concerning zeros of the partition function when generalised to the complex plane of an intensive thermodynamic quantity. Initially, the theory was couched in terms of zeros in the complex fugacity plane which is appropriate for fluids in the grand canonical (fluctuating particle number) ensemble. However, by mapping a lattice gas onto the Ising model [3], the theory was found to hold equally well for a magnet in a fixed magnetic field. Later [4-6] it became clear that the distribution of zeros in the complex temperature plane can reveal information about phase transitions in the canonical ensemble.In section II we present a brief, self-contained discussion of the Lee-Yang theory of equilibrium phase transitions which relates nonanalytic ...

Exact solution of a reaction-diffusion model with particle-number conservation

“…It is predicted that the model has two different phases and in one phase the density profile of the particles has a shock structure. We have confirmed our numerical results by using the Yang-Lee theory of phase transitions [18] which has recently been shown to be applicable to the study of critical behaviors of out-of-equilibrium systems [19,20]. In the present work we will show that by working in the canonical ensemble, the model is exactly solvable in the sense that the thermodynamic limit of physical quantities can be calculated exactly.…”

Information Geometry of Complex Hamiltonians and Exceptional Points

“…In the case of a complex Hamiltonian, however, the associated eigenfunctions need not be analytic in the parameters of the Hamiltonian, and phase transitions can be seen in finite matrix Hamiltonians (see, e.g., [2]). This situation is reminiscent of the analysis proposed by Lee and Yang [3,4], where the breakdown of analyticity associated with the canonical density function in Equation (1) can be explained by extending the parameters into a complex domain (see, e.g., [5] for a heuristic but informative exposition of the Lee-Yang theory). In this case, the canonical density function can exhibit lack of analyticity even in a system with finitely many degrees of freedom, in a way that resembles the eigenstates of finite complex Hamiltonians (see also [6] for a related point of view on these issues).…”

Numerical study of phase transition in an exclusion model with parallel dynamics

“…and therefore the phase transition is of second-order [12]. The canonical partition function (7) as a function of β does not have any real and positive root smaller than one in this case.…”

“…In order to study the phase transitions of the ASEP in canonical ensemble with open boundaries and parallel dynamics, we use the classical Yang-Lee theory by investigating the zeros of (7) in the complex plane of both α and β for different values of L and M. The particle concentration at odd and even sites will also be calculated from (10) and (11) the phase diagram of the model consists of a low-density and a shock phase which are separated by a second-order phase transition at β c = 2ρ. The reason that the phase transition is of a second-order is that the zeros of (7) as a function of β, approach the positive real-β axis at an angle π 4 [12]. The canonical partition function (7) as a function of α, does not have any real and positive root smaller than one in this case.…”