Phase Transition From the Viewpoint of Relaxation Phenomena

We consider a one-dimensional lattice system of unbounded, real-valued spins. We allow arbitrary strong, attractive, nearest-neighbor interaction. We show that the free energy of the canonical ensemble converges uniformly in C 2 to the free energy of the grand canonical ensemble. The error estimates are quantitative. A direct consequence is that the free energy of the canonical ensemble is uniformly strictly convex for large systems. Another consequence is a quantitative local Cramér theorem which yields the strict convexity of the coarse-grained Hamiltonian. With small adaptations, the argument could be generalized to systems with finite-range interaction on a graph, as long as the degree of the graph is uniformly bounded and the associated grand canonical ensemble has uniform decay of correlations.

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“…We start with deducing (107). Let us recall the equations (33) and (53). It follows from a direct computation that…”

Section: 3mentioning

confidence: 99%

Strict Convexity of the Free Energy of the Canonical Ensemble Under Decay of Correlations

Kwon

Menz

2018

J Stat Phys

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“…. , a 2(N −1) ≥ 0 and a 2N > 0, which is used in the study of metastability and phase transitions and may have arbitrarily deep double wells, has been analyzed in [42,44]. Similarly to the experiment in Section 3.5, this example does not satisfy Assumption 2.2 and more stationary states can exist.…”

Section: Nonsymmetric Confining Potentialmentioning

confidence: 99%

Eigenfunction martingale estimators for interacting particle systems and their mean field limit

Pavliotis¹,

Zanoni²

2021

Preprint

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We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, linearized around the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.

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“…• Is for the ce µ m a uniform LSI equivalent to decay of correlations, as it is the case for the gce µ σ (see for example [Yos01,Yos03,HM16])?…”

Section: Let Us Comment On Open Questions and Problemsmentioning

confidence: 99%

“…By the equivalence of dynamic and static phase transitions, a uniform LSI also indicates the absence of a phase transition (see e.g. [Yos03,HM16]). For an introduction to the LSI we refer the reader to [Led01a,Led01b,Roy99,BGL14].…”

Section: Introductionmentioning

confidence: 99%

Uniform LSI for the canonical ensemble on the 1D-lattice with strong, finite-range interaction

Kwon

Menz

2020

ESAIM: PS

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We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show that the canonical ensemble (ce) satisfies a uniform logarithmic Sobolev inequality (LSI). The LSI constant is uniform in the boundary data, the external field and scales optimally in the system size. This extends a classical result of H.T. Yau from discrete to unbounded, real-valued spins. It also extends prior results of Landim, Panizo & Yau or Menz for unbounded, real-valued spins from absent- or weak- to strong-interaction. We deduce the LSI by combining two competing methods, the two-scale approach and the Zegarlinski method. Main ingredients are the strict convexity of the coarse-grained Hamiltonian, the equivalence of ensembles and the decay of correlations in the ce.

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Phase Transition From the Viewpoint of Relaxation Phenomena

Abstract: Some results on the relaxation processes (Glauber dynamics) obtained in the last decade are presented. This article is intended to be a short guided tour through these results for readers without prior knowledge of rigorous statistical mechanics or stochastic processes.

Cited by 10 publications

References 48 publications

Strict Convexity of the Free Energy of the Canonical Ensemble Under Decay of Correlations

Strict Convexity of the Free Energy of the Canonical Ensemble Under Decay of Correlations

Eigenfunction martingale estimators for interacting particle systems and their mean field limit

Uniform LSI for the canonical ensemble on the 1D-lattice with strong, finite-range interaction

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