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

Abstract

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.

Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

“…, where f is defined by (37), M 1 by (18), c by (17) and C is a finite constant depending on γ ∞ , L and the second moment of μ0 and given in (77).…”

“…On the other hand, we prove the existence of multiple invariant probability measures for (6) if the smallness condition on a is not satisfied. Our results for (1) can also be adapted to deal with nonlinear SDEs over the torus T = R/(2πZ), as considered in [18]. As an example, we discuss the application to the Kuramoto model for which a more explicit analysis is available [1,7,8,12].…”

Propagation of chaos: a review of models, methods and applications. II. Applications

“…In the case of the Kuramoto model, this is the law of a highly oscillating system with a frequency of order ε −1 and K = O(ε −1 ). The authors of [DGP21] study a class of McKean-Vlasov gradient systems on the torus which generalizes the Kuramoto model. Using a gradient flow framework (see Section 4.2), one of the main results of the article is an explicit counter example which proves that for some chaotic initial conditions, the two limits N → +∞ and ε → 0 do not commute above the phase transition.…”

Response Theory and Phase Transitions for the Thermodynamic Limit of Interacting Identical Systems

“…For example, for the Kuramoto model of nonlinear oscillators, at the critical noise strength the uniform distribution (on the torus) becomes unstable and stable localized stationary states emerge (phase-locking), leading to synchronization phase transition [14]. A complete theory of phase transitions for the McKean-Vlasov equation on the torus, that includes the Kuramoto model of synchronization, the Hegselmann-Krause model of opinion formation, the Keller-Segel model of chemotaxis etc is presented in [15], see also [16]. The effect of (infinitely) many local minima in the energy landscape on the structure of the bifurcation diagram was studied in [17].…”

“…On the other hand, it has been shown that, for nonlinear oscillators coupled linearly with their mean, the so-called Desai-Zwanzig model [66], the fluctuations at the phase transition point are not Gaussian [1], see also [16] for related results for a variant of the Kuramoto model (the Haken-Kelso-Bunz model). Indeed, the fluctuations are persistent, non-Gaussian in time, with an amplitude described by a nonlinear stochastic differential equation, and associated with a longer timescale [1].…”