We study a model of bacterial dynamics where two interacting random walkers perform run-and-tumble motion on a one-dimensional lattice under mutual exclusion and find an exact expression for the probability distribution in the steady state. This stationary distribution has a rich structure comprising three components: a jammed component, where the particles are adjacent and block each other; an attractive component, where the probability distribution for the distance between particles decays exponentially; and an extended component in which the distance between particles is uniformly distributed. The attraction between the particles is sufficiently strong that even in the limit where continuous space is recovered for a finite system, the two walkers spend a finite fraction of time in a jammed configuration. Our results potentially provide a route to understanding the motility-induced phase separation characteristic of active matter from a microscopic perspective.
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