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We consider a general system of two run-and-tumble particles interacting byhardcore jamming on the unidimensional torus. RTP are a paradigmatic activematter model, typically modeling the evolution of bacteria. By directlymodeling the system at the continuous-space and -time level thanks to piecewisedeterministic Markov processes (PDMP), we derive the conservation conditionswhich sets the invariant distribution and explicitly construct the twouniversality classes for the steady state, the detailed-jamming and theglobal-jamming classes. They respectively identify with the preservation or notof a detailed symmetry at the level of the dynamical internal states betweenprobability flows entering and exiting jamming configurations. Thanks to aspectral analysis of the tumble kernel, we give explicit expressions for theinvariant measure in the general case. The non-equilibrium features exhibitedby the steady state includes positive mass for the jammed configurations and,for the global-jamming class, exponential decay and growth terms, potentiallymodulated by polynomial terms. Interestingly, we find that the invariantmeasure follows, away from jamming configurations, a catenary-like constraint,which results from the interplay between probability conservation and thedynamical skewness introduced by the jamming interactions, seen now as aboundary constraint. This work shows the powerful analytical approach PDMPprovide for the study of the stationary behaviors of RTP sytems and motivatestheir future applications to larger systems, with the goal to derivemicroscopic conditions for motility-induced phase transitions.

Jamming pair of general run-and-tumble particles: Exact results and universality classes