Isothermal Langevin dynamics in systems with power-law spatially-dependent friction

TL;DR

This paper investigates the behavior of Brownian particles in a medium with spatially-dependent diffusion, showing that superdiffusive acceleration predicted by diffusion equations is unphysical and that Langevin dynamics correctly captures the ballistic limit.

Contribution

It demonstrates that superdiffusive acceleration in systems with rapidly decreasing friction is unphysical and confirms Langevin dynamics as the correct approach for such systems.

Findings

PDF matches between analytical and numerical methods at large times.

Superdiffusion occurs for 0<c<1, subdiffusion for c<0.

Superdiffusive acceleration is unphysical, Langevin dynamics shows ballistic limit.

Abstract

We study the dynamics of Brownian particles in a heterogeneous one-dimensional medium with a spatially-dependent diffusion coefficient of the form $D(x)\sim |x|^c$, at constant temperature. The particle's probability distribution function (PDF) is calculated both analytically, by solving Fick's diffusion equation, and from numerical simulations of the underdamped Langevin equation. At large times, the PDFs calculated by both approaches yield identical results, corresponding to subdiffusion for $c<0$, and superdiffusion for $0<c<1$. For $c>1$, the diffusion equation predicts that the particles accelerate. Here, we show that this phenomenon, previously considered in several works as an illustration for the possible dramatic effects of spatially-dependent thermal noise, is unphysical. We argue that in an isothermal medium, the motion cannot exceed the ballistic limit ($\left\langle x^2\right\rangle \sim t^2$). The ballistic limit is reached when the friction coefficient drops sufficiently fast at large distances from the origin, and is correctly captured by Langevin's equation.