Brownian particle in a Poisson-shot-noise active bath: exact statistics, effective temperature, and inference

TL;DR

This paper provides an exact analytical study of a Brownian particle in a Poisson-shot-noise active bath, deriving its stationary distribution, moments, effective temperature, and proposing a method to infer underlying noise parameters from experimental data.

Contribution

It introduces an exact solution for the particle's stationary statistics in a Poisson-shot-noise active environment and develops an inference scheme for experimental analysis.

Findings

Stationary displacement distribution derived analytically.

Effective temperature consistent with equipartition theorem.

Re-entrant transition from non-Gaussian to Gaussian states.

Abstract

We study the dynamics of an overdamped Brownian particle in a thermal bath that contains a dilute solution of active particles. The particle moves in a harmonic potential and experiences Poisson shot-noise kicks with specified amplitude distribution due to moving active particles in the bath. From the Fokker-Planck equation for the particle dynamics we derive the stationary solution for the displacement distribution along with the moments characterizing mean, variance, skewness, and kurtosis, as well as finite time first and second moments. We also compute an effective temperature through the fluctuation-dissipation theorem and show that equipartition theorem holds for all zero-mean kick distributions, including those leading to non-Gaussian stationary statistics. For the case of Gaussian-distributed active kicks we find a re-entrant behaviour from non-Gaussian to Gaussian stationary states and a heavy-tailed leptokurtic distribution across a wide range of parameters as seen in recent experimental studies. Further analysis reveals statistical signatures of the irreversible dynamics of the particle displacement in terms of the time asymmetry of cross-correlation functions. Fruits of our work is the development of a compact inference scheme that may allow experimentalists to extract the rate and moments of underlying shot-noise solely from the statistics of the particle position.