Dynamical transition for a particle in a squared Gaussian potential

TL;DR

This paper investigates the diffusion behavior of a Brownian particle in a Gaussian squared potential, revealing a dynamical transition characterized by a vanishing diffusion constant at a critical temperature, supported by analytical and numerical results.

Contribution

It provides exact diffusion constants in low dimensions, analyzes the dynamical transition, and connects the low-temperature anomalous diffusion to trap model predictions.

Findings

Diffusion constant vanishes at the transition temperature in 1D and 2D.

Finite Fourier modes lead to non-zero low-temperature diffusion with Arrhenius behavior.

Anomalous diffusion exponent matches trap model predictions.

Abstract

We study the problem of a Brownian particle diffusing in finite dimensions in a potential given by $\psi= \phi^2/2$ where $\phi$ is Gaussian random field. Exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power-law fashion at the dynamical transition temperature. Our results are confronted with numerical simulations where the Gaussian field is constructed, in a standard way, as a sum over random Fourier modes. We show that when the number of Fourier modes is finite the low temperature diffusion constant becomes non-zero and has an Arrhenius form. Thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition. In addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model.