Metastability of Nonreversible Random Walks in a Potential Field and the Eyring‐Kramers Transition Rate Formula

Landim, Cláudio; Seo, Il Won

doi:10.1002/cpa.21723

Cited by 35 publications

(71 citation statements)

References 23 publications

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“…By contrast, there is no clear relationship between the capacity and mean jump rate in the non-reversible case. In [3], the collapsed chain is introduced as a potential tool to surmount this difficulty, and this possibility has been confirmed for totally asymmetric zero-range processes [22] and cyclic random walks in a potential field [27].…”

Section: Introductionmentioning

confidence: 89%

“…Recently, a sharp analysis of the capacity for several non-reversible dynamics has also been obtained in [25,27,28], based on the Dirichlet principle [15] and Thomson principle [31] for non-reversible dynamics. These principles are stated in Theorem 5.2.…”

Section: Generalized Dirichlet-thomson Principlesmentioning

confidence: 99%

“…Collapsed chain. The importance of the collapsed chain in the context of metastability has been noticed in [15,3,27]. The collapsed chain can be regarded as a special case of lumped Markov chain (cf.…”

Section: The Same Inequality Holds Formentioning

confidence: 99%

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Condensation of Non-reversible Zero-Range Processes

Seo

2019

Commun. Math. Phys.

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In this article, we investigate the condensation phenomena for a class of nonreversible zero-range processes on a fixed finite set. By establishing a novel inequality bounding the capacity between two sets, and by developing a robust framework to perform quantitative analysis on the metastability of non-reversible processes, we prove that the condensed site of the corresponding zero-range processes approximately behaves as a Markov chain on the underlying graph whose jump rate is proportional to the capacity with respect to the underlying random walk. The results presented in the current paper complete the generalization of the work of Beltran and Landim [4] on reversible zero-range processes, and that of Landim [22] on totally asymmetric zero-range processes on a one-dimensional discrete torus.2000 Mathematics Subject Classification. 60J28, 60K35, 82C22 .

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Section: Introductionmentioning

confidence: 89%

Section: Generalized Dirichlet-thomson Principlesmentioning

confidence: 99%

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Condensation of Non-reversible Zero-Range Processes

Seo

2019

Commun. Math. Phys.

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“…Dynamics which display this behavior are said to "visit points". This class includes condensing zero-range processes [16,84,4,116], random walks in a potential field [91,92,93] or models in which the valleys are singletons as the inclusion process [25] or random walks evolving among random traps [63,62,77,78], but it does not contain the example of Section 1. For such dynamics, in which the entropy plays a role in the metastable behavior, a different approach is needed.…”

Section: Local Ergodicitymentioning

confidence: 99%

“…To obtain sharp bounds, good approximations of the harmonic functions are needed to produce test functions and test flows close to the optimal ones. In concrete examples, one of the difficulties is that the test flows constructed are never divergence free, and a correction has to be introduced to remove the divergence of the test flow, [91,93,116].…”

Section: The Dirichlet and The Thomson Principlesmentioning

confidence: 99%

Metastable Markov chains

Landim¹

2019

Probab. Surveys

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Dirichlet’s and Thomson’s Principles for Non-selfadjoint Elliptic Operators with Application to Non-reversible Metastable Diffusion Processes

Landim

Mariani

Seo

2018

Arch Rational Mech Anal

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We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.When Ω = D \ B, where B ⊂ D ⊂ R d are smooth domains, and f = 1, 0 on B, D c , respectively, the minimal energy is called the capacity. In electrostatics, it corresponds to the total electric charge on the conductor ∂B held at unit potential and grounded at D c . It is denoted by cap D (B) and can be represented, by the divergence theorem, aswhere h is the harmonic function which solves (1.1), n B is the outward normal vector to ∂B, and σ the surface measure at ∂B.where o (1) → 0 as vanishes. This estimate appears in articles published in the 60's. A rigorous proof was first obtained by Bovier, Eckhoff, Gayrard, and Klein [8] with arguments from potential theory, and right after by Helffer, Klein and Nier [14] through Witten Laplacian METASTABILITY IN NON-REVERSIBLE DIFFUSION PROCESSES 3 analysis. We refer to Berglund [6] and Bouchet and Reygner [7] for an historical overview and further references.Recentlty, Bouchet and Reygner [7] extended the Eyring-Kramers formula to the non-reversible setting. They showed that in this context the negative eigenvalue −λ has to be replaced by the unique negative eigenvalue of (Hess U ) (σ) M.We present below a rigorous proof of this result, based on the variational formulae obtained for the capacity in the first part of the article, and on the approach developed by Bovier, Eckhoff, Gayrard, and Klein [8] in the reversible case. This estimate permits to describe the metastable behavior of the diffusion X t in the small noise limit. Analogous results have been derived for random walks in a potential field in [17,18]. Notation and ResultsWe start by introducing the main assumptions. We frequently refer to [12] for results on elliptic equations and to [11,22] for results on diffusions. 2.1. A Dirichlet's and a Thomson's principle. Fix d ≥ 2, and denote by C k (R d ), 0 ≤ k ≤ ∞, the space of real functions on R d whose partial derivatives up to order k are continuous. Let M m,n , 1 ≤ m, n ≤ d, be functions in C 2 (R d ) for which there exists a finite constant C 0 such that d m,n=1 4 C. LANDIM, M. MARIANI, I. SEO Denote by B(r) ⊂ R d , r > 0, the open ball of radius r centered at the origin, and by ∂B(r) its boundary. We assume that lim n→∞ inf z ∈B(n)

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Metastability of Nonreversible Random Walks in a Potential Field and the Eyring‐Kramers Transition Rate Formula

Cited by 35 publications

References 23 publications

Condensation of Non-reversible Zero-Range Processes

Condensation of Non-reversible Zero-Range Processes

Metastable Markov chains

Dirichlet’s and Thomson’s Principles for Non-selfadjoint Elliptic Operators with Application to Non-reversible Metastable Diffusion Processes

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