  We consider N run and tumble particles in one dimension interacting via a
linear 1D Coulomb potential, an active version of the rank diffusion problem.
It was solved previously for N = 2 leading to a stationary bound state in the
attractive case. Here the evolution of the density fields is obtained in the
large N limit in terms of two coupled Burger's type equations. In the
attractive case the exact stationary solution describes a non-trivial
N-particle bound state, which exhibits transitions between a phase where the
density is smooth with infinite support, a phase where the density has finite
support and exhibits "shocks", i.e. clusters of particles, at the edges, and a
fully clustered phase. In presence of an additional linear potential, the phase
diagram, obtained for either sign of the interaction, is even richer, with
additional partially expanding phases, with or without shocks. Finally, a
general self-consistent method is introduced to treat more general
interactions. The predictions are tested through extensive numerical
simulations.
