On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions
arXiv:2001.03920 · doi:10.1007/s00205-021-01648-1
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.
arXiv comment44 pages
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Cited by in corpus (7)
- Propagation of chaos: a review of models, methods and applications. II. Applications
- Perspective: New directions in dynamical density functional theory
- Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions
- Spectroscopy of phase transitions for multiagent systems
- Current fluctuations of a self-interacting diffusion on a ring
- Homogenization of Wasserstein gradient flows
- Entropy maximization in the two-dimensional Euler equations