Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

Bovier, Anton; Hollander, den WThF Frank; Nardi, FR Francesca

doi:10.1007/s00440-005-0460-5

Cited by 56 publications

(125 citation statements)

References 12 publications

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“…The first term contributes an energy −B per particle being present in the system and the second term takes into account some form of nearest neighbor interaction related to 4 An important ingredient of the Hamiltonian (or symplectic) structure is thus lost. In particular the kinematical time-reversal that would normally change the sign of the velocities is absent.…”

Section: Stochastic Lattice Gasesmentioning

confidence: 99%

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A Selection of Nonequilibrium Issues

Maes

́y

Shergelashvili

2009

Lecture Notes in Mathematics

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Abstract:We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetries and inequalities generally valid for nonequilibrium systems, including the fluctuation symmetry, Jarzynski equality, and the direction of currents. In Part II we concentrate on the macroscopic state and how entropy provides a bridge between microscopic dynamics and macroscopic irreversibility; included is a construction of quantum macroscopic states and a result on the equivalence of ensembles.

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Section: Stochastic Lattice Gasesmentioning

confidence: 99%

“…Here we do not consider that. 4 The system thus consists of identical particles that can jump from site to site on the given architecture. The states of the system are assignments to each site of the number of particles.…”

Section: Stochastic Lattice Gasesmentioning

confidence: 99%

A Selection of Nonequilibrium Issues

Maes

́y

Shergelashvili

2009

Lecture Notes in Mathematics

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“…Parts (b)-(e) were proved in [8]. Comparing Theorem 2.2(b) with Theorem 2.1(b), we see that the critical droplet for Kawasaki is more complicated than for Glauber.…”

Section: Kawasaki Dynamicsmentioning

confidence: 79%

Three Lectures on Metastability Under Stochastic Dynamics

Hollander

2009

Lecture Notes in Mathematics

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“…This allowed, for example, to go beyond logarithmic asymptotics for stochastic Ising models in the low-temperature regime ( [21], [29]) and to prove the first rigorous results in the fully conservative case ( [38]), to deal with metastability for the random hopping-time dynamics associated with the Random Energy Model ( [22]), to make a detailed analysis of Sinai's random walk spectrum ( [32]), and to extend the study of the disordered Curie-Weiss model to the case of continuous magnetic field distribution ( [34], [44]). We refer to [48] for a comprehensive account of this approach.…”

Section: A Partial Reviewmentioning

confidence: 99%

Metastable states, quasi-stationary distributions and soft measures

Bianchi

Gaudillière

2016

Stochastic Processes and their Applications

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ABSTRACT. We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypotheses for Markov chains on a finite configuration space in some asymptotic regime. By comparing restricted ensemble and quasi-stationary measures, and introducing soft measures as interpolation between the two, we prove asymptotic exponential exit law and, on a generally different time scale, asymptotic exponential transition law. By using potential-theoretic tools, and introducing "(κ, λ)-capacities", we give sharp estimates on relaxation time, as well as mean exit time and transition time. We also establish local thermalization on shorter time scales. (a) only one thermodynamic phase is present, (b) a system that starts in this state is likely to take a long time to get out, (c) once the system has gotten out, it is unlikely to return. We can think, for example, about freezing fog made of small droplets in which only one phase is present (liquid phase) that remains for a long time in such a state (until collision with ground or trees, forming then hard rime) and that once frozen will typically not return to liquid state before pressure or temperature have changed.To model such a state they considered in [4] a deterministic dynamics with equilibrium measure µ. First, they associated with the metastable phase a subset R of the configuration space, and described this metastable state by the restricted ensemble µ R = µ(·|R). Second, they proved that the escape rate from R of the system started in µ R is maximal at time t = 0, and that this initial escape rate is very small. Last, they used standard methods of equilibrium statistical mechanics to deal with (c). As an estimate of the returning probability to the metastable state they used the fraction of members of the equilibrium ensemble that have configurations in R and they noted ([4], Section 8):This amounts to assuming that a system whose dynamical state has just left R is no more likely to return to it than one whose dynamical state was never anywhere near R. The validity of this assumption, at least in the short run, is dubious, but at least it provides us with some indication of what to expect. In this paper we want to give a different model for the same phenomenology that overcomes the last difficulty. We will work with stochastic processes rather than deterministic dynamics, but the Lebowitz-Penrose modelization will be our guideline. We will try to recover this phenomenology under simple and practical hypotheses only. Since the study of metastability has been considerably enriched after the Lebowitz and Penrose work, we want also to incorporate in our modeling as much as possible of what was previously achieved. We will then make a brief and partial review of these achievements. Our goals and starting ideas will depend on this review but not our proofs, since we want to make this paper as self-contained as possible. Our model and results are presented in Section 2. Examples of applications are given in Section 7.

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Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

Cited by 56 publications

References 12 publications

A Selection of Nonequilibrium Issues

A Selection of Nonequilibrium Issues

Three Lectures on Metastability Under Stochastic Dynamics

Metastable states, quasi-stationary distributions and soft measures

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