Steady state and mixing of two run-and-tumble particles interacting through jamming and attractive forces

TL;DR

This work analyzes two interacting run-and-tumble particles on the real line with an attractive potential and jamming that prevents crossing, focusing on three PDMP-driven models: instantaneous linear, finite linear, and instantaneous harmonic. It delivers explicit invariant measures for all models, revealing Dirac masses at the boundary and regime-dependent bulk densities, illustrating a near- vs far-from-equilibrium classification. The authors develop non-asymptotic convergence bounds using coupling techniques: exponential mixing in total variation for the linear models and exponential Wasserstein contraction for the harmonic model, with sharp rate characterizations in terms of model parameters. Together, these results illuminate clustering phenomena, long-time behavior, and the speed of relaxation toward steady states in a minimal active-particle system with boundary constraints, offering insights applicable to broader active-matter contexts.

Abstract

We study the long-time behavior of two run-and-tumble particles on the real line subjected to an attractive interaction potential and jamming interactions, which prevent the particles from crossing. We provide the explicit invariant measure, a useful tool for studying clustering phenomena in out-ofequilibrium statistical mechanics, for different tumbling mechanisms and potentials. An important difference with invariant measures of equilibrium systems are Dirac masses on the boundary of the state space, due to the jamming interactions. Qualitative changes in the invariant measure depending on model parameters are also observed, suggesting, like a growing body of evidence, that run-andtumble particle systems can be classified into close-to-equilibrium and strongly out-of-equilibrium models. We also study the relaxation properties of the system, which are linked to the timescale at which clustering emerges from an arbitrary initial configuration. When the interaction potential is linear, we show that the total variation distance to the invariant measure decays exponentially and provide sharp bounds on the decay rate. When the interaction potential is harmonic, we give quantitative exponential bounds in a Wasserstein-type distance.

Figures (4)

• Figure 1: Single-particle velocity transition rates
• Figure 2: Relative velocity transition rates
• Figure 3: Realization of the relative distance $x$
• Figure 4: Single velocity coupling

Theorems & Definitions (52)

• Definition 1: Interacting run-and-tumble process
• Proposition 3
• Theorem 4
• Theorem 5
• Proposition 6: Invariant measure of the instantaneous linear process
• proof
• Lemma 7
• Lemma 9
• Proposition 10: Invariant measure of the finite linear process
• proof : Proof of Proposition \ref{['prop:finite_linear_invariant_measure']}