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

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.

10 page, 9 figures