  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. L\'{e}vy 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 L\'{e}vy processes. In the
infinite case we show that the gap process of the particle system on the
half-line is tight for appropriate initial conditions and same drift and
diffusion coefficients for all particles. Applications of such processes
include the modeling of capital distributions among the ranked participants in
a financial market, the stability of certain stochastic queueing and storage
networks and the study of the Sherrington--Kirkpatrick model of spin glasses.
