Yang$ndash$Lee zeros for a nonequilibrium phase transition

Dammer, Stephan M.; Dahmen, Sílvio R.; Hinrichsen, Haye

doi:10.1088/0305-4470/35/21/303

Cited by 40 publications

(36 citation statements)

References 18 publications

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“…As is evident from Fig. 7, which was presented in [48], the zeros do not lie along a single curve but along a sequence of curves meeting at the critical point (as well as inside some region that encloses part of the real axis for p > 2).…”

Section: Iv2 Reaction-diffusion Systems and Directed Percolationmentioning

confidence: 88%

“…Otherwise, on a finite system the steady state is simply the absorbing state and the steady-state normalisation is trivially equal to a constant. Nevertheless the work of [47,48], which we reviewed in section IV.2, indicates that the Lee-Yang theory can be relevant when one considers other properties of the nonequilibrium system such as the percolation probability.…”

Section: Discussionmentioning

confidence: 99%

“…In the work of [48], numerical solutions for small systems were used successfully to estimate the transition point and density of zeros as it is approached. From this information one learns about the nature of the phase transition and, for example, can estimate the values of critical exponents.…”

Section: Discussionmentioning

confidence: 99%

“…Recently an attempt has been made to shed further light on the DP transition by studying the zeros of the percolation probability P n (p) [47,48]. At first glance, a connection between this probability and a partition function is not obvious.…”

Section: Iv2 Reaction-diffusion Systems and Directed Percolationmentioning

confidence: 99%

“…Zeros of the percolation probability on the directed percolation lattice up to depths n = 15. Taken from [48] in which the symbol t is used where we use n.…”

Section: Iv2 Reaction-diffusion Systems and Directed Percolationmentioning

confidence: 99%

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The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

2003

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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 ...

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Section: Iv2 Reaction-diffusion Systems and Directed Percolationmentioning

confidence: 88%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Iv2 Reaction-diffusion Systems and Directed Percolationmentioning

confidence: 99%

“…Zeros of the percolation probability on the directed percolation lattice up to depths n = 15. Taken from [48] in which the symbol t is used where we use n.…”

Section: Iv2 Reaction-diffusion Systems and Directed Percolationmentioning

confidence: 99%

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The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

2003

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Demagnetization via Nucleation of the Nonequilibrium Metastable Phase in a Model of Disorder

2008

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We study both analytically and numerically demagnetization via nucleation of the metastable phase in a two-dimensional nonequilibrium Ising ferromagnet at temperature T . Canonical equilibrium is dynamically impeded by a weak random perturbation which models homogeneous disorder of undetermined source. We present a simple theoretical description, in perfect agreement with Monte Carlo simulations, assuming that the decay of the nonequilibrium metastable state is due, as in equilibrium, to the competition between the surface and the bulk. This suggests one to accept a nonequilibrium free-energy at a mesoscopic/cluster level, and it ensues a nonequilibrium surface tension with some peculiar low-T behavior. We illustrate the occurrence of intriguing nonequilibrium phenomena, including: (i) stochastic resonance phenomena at low T which stabilize the metastable state as temperature increases; (ii) reentrance of the limit of metastability under strong nonequilibrium conditions; and (iii) noise-enhanced propagation of domain walls. The cooperative behavior of our system, which is associated to the interplay between thermal and nonequilibrium fluctuations, may also be understood in terms of a Langevin equation with additive and multiplicative noises. We also studied metastability in the case of open boundaries as it may correspond to a magnetic nanoparticle. We then observe the most irregular relaxation triggered by the additional surface randomness. In particular, at low T, the relaxation becomes discontinuous as occurring by way of scale-free avalanches, so that it resembles the type of relaxation reported for many complex systems. We show that this results from the superposition of many demagnetization events, each with a well-defined scale which is determined by the curvature of the domain wall at which it originates. This is an example of (apparent) scale invariance in a nonequilibrium setting which is not to be associated with any familiar kind of criticality. a metastable state in a properly constrained (equilibrium) ensemble (2; 3; 9), and that most equilibrium concepts may easily be adapted (4; 5; 6; 7; 8).In this paper we present a detailed study of metastability (and nucleation) in a nonequilibrium model. In order to deal with a simple microscopic model of metastability, we study a two-dimensional kinetic Ising system, as in previous studies. However, for the system to exhibit nonequilibrium behavior, time evolution is defined here as a superposition of the familiar thermal process at temperature T and a weak completely-random process. This competition is probably one of the simplest, both conceptually and operationally, ways of impeding equilibrium. Furthermore, one may argue that it captures some underlying disorder induced by random impurities or other causes which are unavoidable in actual samples. The specific origin for such dynamic randomness will vary with the situation considered. We mention that a similar mechanism has already been used to model the macroscopic consequences of rapidly-diffusing local ...

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