Stationary States in Driven Collisional Systems with Application to

We study a driven particle system whose velocity distribution f(v; t) satisses a Boltzmann equation with a nonlinear collision term, and linear terms representing collisions with thermalized particles of another species having a speciied Maxwellian distribution, and a driving force. We prove that when the non-linear terms dominate, f(v; t) is kept close to a Maxwellian distribution M(v; u(t); e(t)) with parameters u(t) and e(t) satisfying a system of nonlinear equations { the \hydrodynamic" equations. This result holds even when their stationary solution is non-unique, corresponding to a dynamical phase transition for f in such systems. We apply our results to a model of a partially ionized spatially homogeneous plasma in an external eld E.