Jamming pair of general run-and-tumble particles: Exact results, symmetries and steady-state universality classes

Abstract

While run-and-tumble particles are a foundational model for self-propelled particles as bacteria or Janus particles, the analytical derivation of their steady state from the microscopic details is still an open problem. By directly modeling the system at the continuous-space and -time level thanks to piecewise deterministic Markov processes (PDMP), we derive the conservation conditions which sets the invariant distribution and, more importantly, explicitly construct the two universality classes for the steady state, the detailed-jamming and the global-jamming classes. They respectively identify with the preservation or not in a detailed manner of a symmetry at the level of the dynamical internal states between probability flows entering and exiting jamming configurations. We call such symmetry active global balance, as it is the true nonequilibrium counterpart of the equilibrium global balance. Thanks to a spectral analysis of the tumble kernel, we give explicit expressions for the invariant measure in the general case. We show that the non-equilibrium features exhibited by the steady state include positive mass for the jammed configurations and, for the global-jamming class, exponential decay and growth terms, potentially modulated by polynomial terms. Interestingly, we find that the invariant measure follows, away from jamming configurations, a catenary-like constraint, which results from the interplay between probability conservation and the dynamical skewness introduced by the jamming interactions, seen now as a boundary constraint. This work shows the powerful analytical approach PDMP provide for the study of the stationary behaviors of RTP systems and motivates their future applications to larger systems, with the goal to derive microscopic conditions for motility-induced phase transitions.