Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

Redig, Frank; Wang, Feijia

doi:10.1007/s10955-012-0523-9

Cited by 10 publications

(13 citation statements)

References 16 publications

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“…Example: Two state symmetric flipping To see a concrete example of an explicit solution, we consider the case of two states flipping at rate 1, which corresponds with mean-field independent spin flip dynamics, treated before in [12], [4], [14].…”

Section: Hamiltonian Trajectories For Finite Markov Chainsmentioning

confidence: 99%

“…In this paper we focus on the computation of the lagrangian L with the scheme of Feng and Kurtz [6], explained e.g. in [14].…”

Section: Introductionmentioning

confidence: 99%

“…In [4] we explained how Gibbs-non-Gibbs transitions in lattice spin systems can be related to a bifurcation phenomenon for the optimal trajectories -in the sense of large deviations-of the empirical measure conditioned to arrive at a given measure at a fixed time T > 0. In [12], [14] this idea was developed in the mean-field context, i.e., for the trajectory of the magnetization. The idea is to consider a translation invariant stochastic dynamics, and study the trajectory of the empirical measure.…”

Section: Introductionmentioning

confidence: 99%

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Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

Redig

Wang

2014

J Stat Phys

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We compute the Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) as well as diffusion processes. For diffusion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy or relative entropy density per unit time.

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Section: Hamiltonian Trajectories For Finite Markov Chainsmentioning

confidence: 99%

“…In this paper we focus on the computation of the lagrangian L with the scheme of Feng and Kurtz [6], explained e.g. in [14].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

Redig

Wang

2014

J Stat Phys

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“…Proof of Theorem 1.5: We use the Feng-Kurtz scheme as presented in [16,11,36] to show convergence on the level of trajectories. The Feng-Kurtz Hamiltonian for the generator Q emp N reads…”

Section: Lemma 22mentioning

confidence: 99%

Synchronization for discrete mean-field rotators

Jahnel

Külske

2014

Electron. J. Probab.

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We analyze a non-reversible mean-field jump dynamics for discrete qvalued rotators and show in particular that it exhibits synchronization. The dynamics is the mean-field analogue of the lattice dynamics investigated by the same authors in [26] which provides an example of a non-ergodic interacting particle system on the basis of a mechanism suggested by Maes and Shlosman [32].Based on the correspondence to an underlying model of continuous rotators via a discretization transformation we show the existence of a locally attractive periodic orbit of rotating measures. We also discuss global attractivity, using a free energy as a Lyapunov function and the linearization of the ODE which describes typical behavior of the empirical distribution vector.

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“…instead of (1.9) (cf. [17,Eq. (25)]), and so we obtain completely analogous results (in Corollary 1.12 the condition Φ 2 V > − 1+t 2t is replaced by Φ 2 V > −(e 2t − 1) −1 ).…”

mentioning

confidence: 99%

Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions

Hollander

Redig

Zuijlen

2015

Stochastic Processes and their Applications

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We consider a system of real-valued spins interacting with each other through a meanfield Hamiltonian that depends on the empirical magnetisation of the spins. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. Using large deviation theory we show that there exists an explicitly computable crossover time t c ∈ [0, ∞] from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness (t c = 0), short-time conservation and large-time loss of Gibbsianness (t c ∈ (0, ∞)), and preservation of Gibbsianness (t c = ∞). Depending on the potential, the system can be Gibbs or non-Gibbs at the crossover time t = t c .MSC 2010. 60F10, 60K35, 82C22, 82C27.

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Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

Cited by 10 publications

References 16 publications

Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

Synchronization for discrete mean-field rotators

Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions

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