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

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-of-equilibrium 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-and-tumble 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.