Activity-driven clustering and many-body steady state of jamming run-and-tumble particles

We exactly resolve the three-particle steady state of run-and-tumble particles with jamming interactions, providing the first microscopic description beyond two bodies. The invariant measure, derived via a piecewise-deterministic Markov process description and symmetry principles, reveals persistent, separated, and diffusive regimes ruled by the activity parameter. A geometric cascade of scales in the activity parameter organizes the structural weights, showing the separated phase dominates at finite activity, while non-uniformity plays only a minor role. Extending these results to larger systems, we show that the $N$-body steady state inherits the same organization: the number of clusters becomes sharply defined by the activity value, with crossover boundaries whose slopes diverge with $N$. We also show how the activity plays a role similar to a fugacity conjugate to cluster number, yielding a grand-canonical-like structure emerging directly from the microscopic dynamics. This framework lays the groundwork for a systematic microscopic theory of active many-body steady states.