  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.
