A General Approach to the Shape Transition of Run-and-Tumble Particles: The 1D PDMP Framework for Invariant Measure Regularity

TL;DR

This paper investigates the shape transition in the invariant measure of 1D run-and-tumble particles under potential, revealing how regularity and divergence depend on tumble rates, and extends the mathematical theory of invariant measure regularity.

Contribution

It provides a general framework for analyzing the shape transition of RTPs' invariant measures without explicit solutions, extending regularity theory for 1D systems with random switching.

Findings

Invariant measure density is continuous at high tumble rates.

Divergences in density occur at low tumble rates, indicating shape transition.

The analysis confirms shape transition as a robust feature of RTPs under potential.

Abstract

Run-and-tumble particles (RTPs) have emerged as a paradigmatic example for studying nonequilibrium phenomena in statistical mechanics. The invariant measure of a wide class of RTPs subjected to a potential possesses a density that is continuous at high tumble rates but exhibits divergences at low ones. This key feature, known as shape transition, constitutes a qualitative indicator of the relative closeness (continuous density) or strong deviation (diverging density) from the equilibrium setting. Furthermore, the points at which the density diverges correspond to the configurations where the system spends most of its time in the low tumble rate regime. Building on and extending existing results concerning the regularity of the invariant measure of one-dimensional dynamical systems with random switching, we show how to characterize the shape transition even in situations where the invariant measure cannot be computed explicitly. Our analysis confirms shape transition as a robust, general feature of RTPs subjected to a potential. We also refine the regularity theory for the invariant measure of one-dimensional dynamical systems with random switching.