Gap statistics of two interacting run and tumble particles in one dimension

TL;DR

This paper analyzes the steady-state distribution and relaxation dynamics of the gap between two interacting run and tumble particles in one dimension, considering thermal noise and different geometries, revealing exponential localization and spectral properties.

Contribution

It provides an analytical computation of the gap distribution, explores the spectral classification of the evolution operator, and characterizes the crossover in relaxation time with system size.

Findings

Steady-state gap distribution is exponentially localized in space.

Relaxation time exhibits a crossover from size-independent to size-dependent behavior.

Eigenvalue spectrum classified into four symmetry sectors with explicit low-lying eigenvalues for large systems.

Abstract

We study the dynamics of the separation (gap) between a pair of interacting run and tumble particles (RTPs) moving in one dimension in the presence of additional thermal noise. On a ring geometry the distribution of the gap approaches a steady state. We analytically compute this distribution and find that this is exponentially localised in space, in contrast to the `jammed' configuration, seen earlier in the absence of thermal noise. We also study the relaxation which is an exponential, characterised by a time scale $\tau_r$. We observe that this time scale undergoes a crossover from a size independent value to a size dependent form with increasing size $l$ of the ring. We study the full eigenvalue spectrum of the evolution operator $\mathcal{L}$ and find that the spectrum can be classified into four sectors depending on the symmetries of $\mathcal{L}$. For large $l$, we find explicit expressions for the low lying eigenvalues in each of the sectors. On infinite line the separation does not reach a steady state. In the long times we find that the particles behave as interacting Brownian particles, except for the presence of a peak in the distribution at small separation which is a remnant of activity.