Equilibrium large deviations for mean-field systems with translation invariance

Reygner, Julien

doi:10.1214/17-aap1379

“…Indeed, applying Lemma 1.8, one can show that the N -particle Gibbs measure M N converges, in the sense of Definitions 1.1 and 1.2, to X ∈ P(P(T)), where X is supported uniformly on the set of translates ofν min β , see [35,Proposition 3]. The alternative is to work in a quotient space as in [34,39] or to add a small confinement to break the translation invariance of the problem. We choose to do the latter.…”

Section: The Effect Of Phase Transitionsmentioning

confidence: 99%

On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

Delgadino

¹

,

Gvalani

²

,

Pavliotis

³

2021

Arch Rational Mech Anal

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The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.

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“…Indeed, applying Lemma 1.3, one can show that the N -particle Gibbs measure M N converges, in the sense of Definitions 1.1 and 1.2, to X ∈ P(P(T)), where X is supported uniformly on the set of translates of νmin β . The alternative is to work in a quotient space as in [38,34] or to add a small confinement to break the translation invariance of the problem. We choose to do the latter.…”

Section: ŵ (K) :=mentioning

confidence: 99%

On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions

Delgadino,

Gvalani,

Pavliotis

2020

Preprint

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The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature. Contents1. Introduction 1.1. Overview 1.2. Set up and preliminaries 1.3. The space P(P(R d )) as the limit of P sym (R d ) N 1.4. Gradient flow formulation and the mean field limit 1.5. Scaling and the quotiented process 1.6. The diffusive limit 1.7. The limit N → ∞ followed by ε → 0 1.8. The limit ε → 0 followed by N → ∞ 1.9. The effect of phase transitions 1.10. Application of the fluctuation theorem 2. Proof of Theorem 1.5 3. Proof of Theorem 1.7 4. Proofs of Section 1.9 Appendix A. Coupling arguments References

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“…The LDP with speed N for the empirical density corresponding to the Dean equation (2.13) on the time window [0, T ], was obtained by Dawson and Gärtner in the case R = ∞ [28,29], and is given for any function f := f (q, θ , t) by [92,30,71,31,32,26,55,18,54,4,17,78,86,50,24] ǫ…”

Section: Kinetic Large Deviation Functionalmentioning

confidence: 99%

“…The dynamics becomes thus infinite-dimensional and the typical behavior of f ǫ N (q, θ , t) is described by f ǫ (q, θ , t) which is solution of a (kind of) McKean-Vlasov equation [75,76,44,70,27,59,79,12,60,91,77,19]. Fluctuations (central limit theorems or large deviations principles) around this typical behavior have been investigated previously [92,30,71,31,32,26,55,18,54,4,17,78,86,50,24]. More explicitly a large deviations principle for f ǫ N holds 3 :…”

mentioning

confidence: 99%

Gamma Convergence Approach For The Large Deviations Of The Density In Systems Of Interacting Diffusion Processes

Barré,

Bernardin,

Chétrite

et al. 2019

Preprint

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We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean-Vlasov equation depending on ǫ, the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle (LDP) with a rate functional ǫ . We study the Γ -convergence of ǫ as ǫ → 0 and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory (MFT) for diffusive systems.

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Equilibrium large deviations for mean-field systems with translation invariance

Cited by 7 publications

References 48 publications

On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions

Gamma Convergence Approach For The Large Deviations Of The Density In Systems Of Interacting Diffusion Processes

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