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

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 piecewise-deterministic Markov processes (PDMPs), 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 PDMPs.