Exact moments for a run and tumble particle in a harmonic trap with a finite tumble time

TL;DR

This paper derives exact moments and distributions for a run and tumble particle in a harmonic trap with finite tumble time, providing analytical tools for understanding its steady state and dynamics.

Contribution

It introduces a novel exact analytical framework for moments and distributions of run and tumble particles with finite tumble times in harmonic traps.

Findings

Exact steady state distribution derived

Programmable Volterra difference equation established

Moments used to infer distribution via Gaussian quadrature

Abstract

We study the problem of a run and tumble particle in a harmonic trap, with a finite run and tumble time, by a direct integration of the equation of motion. An exact 1D steady state distribution, diagram laws and a programmable Volterra difference equation are derived to calculate any order of moments in any other dimension, both for steady state as well as the Laplace transform in time for the intermediate states. We also use the moments to infer the distribution by considering a Gaussian quadrature for the corresponding measure, and from the scaling law of high order moments.