Diffusion caused by two noises-active and thermal

TL;DR

This paper models the diffusion of colloids in active systems using an overdamped Langevin equation with both thermal and active noise, deriving the propagator and analyzing how different active noises affect distribution dynamics.

Contribution

It introduces a phase-space path integral method to derive the propagator and moments for systems driven by various active noise models, highlighting deviations from Gaussian behavior.

Findings

PDF deviates from Gaussian at short times for non-Gaussian noises

Distribution converges to Gaussian at long times due to the central limit theorem

Different active noises produce distinct short-time distribution features

Abstract

The diffusion of colloids inside an active system-e.g. within a living cell or the dynamics of active particles itself (e.g. self-propelled particles) can be modeled through overdamped Langevin equation which contains an additional noise term apart from the usual white Gaussian noise, originating from the thermal environment. The second noise is referred to as 'active noise' as it arises from activity such as chemical reactions. The probability distribution function (PDF or the propagator) in space-time along with moments provides essential information for understanding their dynamical behavior. Here we employ the phase-space path integral method to obtain the propagator, thereby moments and PDF for some possible models for such noise. At first, we discuss the diffusion of a free particle driven by active noise. We consider four different possible models for active noise, to capture the possible traits of such systems. We show that the PDF for systems driven by noises other than Gaussian noise largely deviates from normal distribution at short to intermediate time scales as a manifestation of out-of-equilibrium state, albeit converges to Gaussian distribution after a long time as a consequence of the central limit theorem. We extend our work to the case of a particle trapped in a harmonic potential and show that the system attains steady state at long time limit. Also, at short time scales, the nature of distribution is different for different noises, e.g. for particle driven by dichotomous noise, the probability is mostly concentrated near the boundaries whereas a long exponential tail is observed for a particle driven by Poissonian white noise.