We consider finite and infinite systems of particles on the real line and
half-line evolving in continuous time. Hereby, the particles are driven by
i.i.d. Levy processes endowed with rank-dependent drift and diffusion
coefficients. In the finite systems we show that the processes of gaps in the
respective particle configurations possess unique invariant distributions and
prove the convergence of the gap processes to the latter in the total variation
distance, assuming a bound on the jumps of the Levy processes.