Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Luçon, Éric

doi:10.1016/j.jfa.2014.03.008

Cited by 4 publications

(7 citation statements)

References 38 publications

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“…Namely, the purpose of previous work [28] was to prove a quenched fluctuation result for the empirical measure (1.2) around its mean-field limit (1.3) on a finite time horizon [0, T ]. The main conclusion of [28] was that these fluctuations are disorder dependent and the long time analysis of the limiting fluctuations [30] suggested a non-self-averaging phenomenon for (1.1) similar to the one observed here.…”

Section: Links With Existing Modelssupporting

confidence: 75%

Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model

Luçon¹,

Poquet²

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. The aim of the paper is to address the long time behavior of the Kuramoto model of mean-field coupled phase rotators, subject to white noise and quenched frequencies. We analyse the influence of the fluctuations of both thermal noise and frequencies (seen as a disorder) on a large but finite population of N rotators, in the case where the law of the disorder is symmetric. On a finite time scale [0, T ], the system is known to be self-averaging: the empirical measure of the system converges as N → ∞ to the deterministic solution of a nonlinear Fokker-Planck equation which exhibits a stable manifold of synchronized stationary profiles for large interaction. On longer time scales, competition between the finite-size effects of the noise and disorder makes the system deviate from this mean-field behavior. In the main result of the paper we show that on a time scale of order √ N the fluctuations of the disorder prevail over the fluctuations of the noise: we establish the existence of disorder-induced traveling waves for the empirical measure along the stationary manifold. This result is proved for fixed realizations of the disorder and emphasis is put on the influence of the asymmetry of these quenched frequencies on the direction and speed of rotation of the system. Asymptotics on the drift are provided in the limit of small disorder.

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Section: Links With Existing Modelssupporting

confidence: 75%

Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model

Luçon¹,

Poquet²

2017

Ann. Inst. H. Poincaré Probab. Statist.

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“…The question is natural because for the limit PDE [13,26] there is a contractive manifold similar to M [20]. However the results in [27] suggest that a nontrivial dynamics on the contractive manifold is observed rather on times proportional to √ N and one expects a dynamics with a nontrivial random drift. The role of disorder in this type of models is not fully elucidated (see however [12] on the critical case) and the global long time dynamics represents a challenging issue.…”

Section: 5mentioning

confidence: 99%

Synchronization and random long time dynamics for mean-field plane rotators

Bertini

Giacomin

Poquet

2013

Probab. Theory Relat. Fields

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We consider the natural Langevin dynamics which is reversible with respect to the mean-field plane rotator (or classical spin XY) measure. It is well known that this model exhibits a phase transition at a critical value of the interaction strength parameter {Mathematical expression}, in the limit of the number {Mathematical expression} of rotators going to infinity. A Fokker-Planck PDE captures the evolution of the empirical measure of the system as {Mathematical expression}, at least for finite times and when the empirical measure of the system at time zero satisfies a law of large numbers. The phase transition is reflected in the fact that the PDE for {Mathematical expression} above the critical value has several stationary solutions, notably a stable manifold-in fact, a circle-of stationary solutions that are equivalent up to rotations. These stationary solutions are actually unimodal densities parametrized by the position of their maximum (the synchronization phase or center). We characterize the dynamics on times of order {Mathematical expression} and we show substantial deviations from the behavior of the solutions of the PDE. In fact, if the empirical measure at time zero converges as {Mathematical expression} to a probability measure (which is away from a thin set that we characterize) and if time is speeded up by {Mathematical expression}, the empirical measure reaches almost instantaneously a small neighborhood of the stable manifold, to which it then sticks and on which a non-trivial random dynamics takes place. In fact the synchronization center performs a Brownian motion with a diffusion coefficient that we compute. Our approach therefore provides, for one of the basic statistical mechanics systems with continuum symmetry, a detailed characterization of the macroscopic deviations from the large scale limit-or law of large numbers-due to finite size effects. But the interest for this model goes beyond statistical mechanics, since it plays a central role in a variety of scientific domains in which one aims at understanding synchronization phenomena. © 2013 Springer-Verlag Berlin Heidelberg

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“…In this paper, we review some recent results ( [39,40,41]) concerning large population asymptotics of interacting diffusions in a random environment. This class of models generalizes systems of particles in a mean-field interaction that have been intensively studied since McKean [44].…”

Section: Diffusions In Mean-field Interactionmentioning

confidence: 99%

“…the disorder) to the process C with covariance (21). In the framework of the Kuramoto model with binary disorder (recall Section 2.2.1), computations show ( [40]) that the relevant quantity for the dynamics of (22) is the restriction C + of the process C to the component on +1:…”

Section: Quenched Central Limit Theoremmentioning

confidence: 99%

Large Population Asymptotics for Interacting Diffusions in a Quenched Random Environment

Luçon

2015

Springer Proceedings in Mathematics &Amp; Statistics

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We review some recent results on large population behavior of interacting diffusions in a random environment. Emphasis is put on the quenched influence of the environment on the macroscopic behavior of the system (law of large numbers and fluctuations). We address the notion of (non-)self-averaging phenomenon for this class of models. A guiding thread in this survey is the Kuramoto synchronization model which has met in recent years a growing interest in the literature.

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Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Cited by 4 publications

References 38 publications

Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model

Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model

Synchronization and random long time dynamics for mean-field plane rotators

Large Population Asymptotics for Interacting Diffusions in a Quenched Random Environment

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