Dynamics of the Desai-Zwanzig model in multi-well and random energy landscapes

TL;DR

This paper investigates the stationary states, bifurcations, and phase transitions of a mean field model with weakly interacting diffusions in multi-well and random energy landscapes, using analytical and extensive numerical methods.

Contribution

It provides a detailed analysis of bifurcations and phase transitions in a generalized Desai-Zwanzig model with both deterministic and random potential minima, combining analytical solutions and numerical simulations.

Findings

Characterized the structure of bifurcations in the model.

Identified phase transition phenomena in multi-well landscapes.

Demonstrated the effects of randomness in potential minima on system behavior.

Abstract

We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {\bf 19}1-24 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multi-well potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional and a continuation algorithm for the stationary solutions.