Noise-induced transitions in rugged energy landscapes

TL;DR

This paper analyzes how multiscale rugged energy landscapes influence the stochastic dynamics of particles, revealing noise-induced transitions, hysteresis effects, and deriving explicit formulas for effective diffusion in complex potentials.

Contribution

It introduces a coarse-grained Langevin model for multiscale potentials and provides explicit formulas and bifurcation analysis for noise-induced phenomena.

Findings

Multiscale potentials cause hysteresis and noise-induced transitions.

Explicit formula for effective diffusion in self-similar potentials.

Analysis of the limit as the number of small scales approaches infinity.

Abstract

We consider the problem of an overdamped Brownian particle moving in multiscale potential with N + 1 characteristic length scales: the macroscale and N separated microscales. We show that the coarse-grained dynamics is given by an overdamped Langevin equation with respect to the free energy and with a space dependent diffusion tensor, the calculation of which requires the solution of N fully coupled Poisson equations. We study in detail the structure of the bifurcation diagram for one-dimensional problems and we show that the multiscale structure in the potential leads to hysteresis effects and to noise-induced transitions. Furthermore, we obtain an explicit formula for the effective diffusion coefficient for a self-similar separable potential and we investigate the limit of infinitely many small scales.