Active noise-driven particles under space-dependent friction in one dimension

TL;DR

This paper analyzes a one-dimensional active particle model with space-dependent friction and potential, providing analytical and numerical insights into its short-time dynamics and stationary distributions, which vary with system parameters.

Contribution

It introduces a comprehensive study of active particles with space-dependent friction and potential, revealing how their stationary PDFs transition between different distributions based on parameter tuning.

Findings

Short-time dynamics show diffusive and ballistic regimes.

Stationary PDFs can be Laplacian, Gaussian, or bimodal.

Distribution shapes depend on friction and potential exponents.

Abstract

We study a Langevin equation describing the stochastic motion of a particle in one dimension with coordinate $x$, which is simultaneously exposed to a space-dependent friction coefficient $\gamma(x)$, a confining potential $U(x)$ and non-equilibrium (i.e., active) noise. Specifically, we consider frictions $\gamma(x)=\gamma_0 + \gamma_1 | x |^p$ and potentials $U(x) \propto | x |^n$ with exponents $p = 1,2$ and $n = 0, 1, 2$. We provide analytical and numerical results for the particle dynamics for short times and the stationary probability density functions (PDFs) for long times. The short-time behaviour displays diffusive and ballistic regimes while the stationary PDFs display unique characteristic features depending on the exponent values $(p, n)$. The PDFs interpolate between Laplacian, Gaussian and bimodal distributions, whereby a change between these different behaviours can be achieved by a tuning of the friction strengths ratio $\gamma_0/\gamma_1$. Our model is relevant for molecular motors moving on a one-dimensional track and can also be realized for confined self-propelled colloidal particles.