Diffusing diffusivity model with dichotomous noise

TL;DR

This paper models Langevin dynamics with stochastic diffusivity driven by dichotomous noise, deriving analytical expressions for particle displacement PDF and analyzing its asymptotic behavior, revealing a transition to Gaussian diffusion at long times.

Contribution

It introduces a minimal, analytically tractable model of stochastic transport with bounded diffusivity driven by dichotomous noise, providing new insights into short- and long-time behaviors.

Findings

Short-time PDF exhibits logarithmic divergence at the origin.

Tails of the PDF transition from exponential to Gaussian modulated by a power-law.

Long-time dynamics converges to ordinary Gaussian diffusion.

Abstract

We study Langevin dynamics with stochastic diffusivity arising from fluctuations of the surrounding medium. The diffusivity is modeled as Ornstein-Uhlenbeck process driven by symmetric dichotomous noise, which confines it to a finite interval. We derive analytical expressions for the short-time probability density function (PDF) of the particle displacement and analyse its asymptotic behaviour. While the PDF retains the characteristic logarithmic divergence at the origin, its tails differ from the Gaussian white-noise case: exponential tails are replaced by Gaussian ones modulated by a power-law with a switching-rate-dependent exponent. At long times, the dynamics converges to ordinary Gaussian diffusion. We determine the variance and covariance of the time-averaged stochastic diffusivity and show that it is self-averaging. The model provides a minimal analytically tractable framework for stochastic transport in environments with bounded or switching fluctuations.