  We consider a charged Brownian gas under the influence of external and non
uniform electric, magnetic and mechanical fields, immersed in a non uniform
bath temperature. With the collision time as an expansion parameter, we study
the solution to the associated Kramers equation, including a linear reactive
term. To first order we obtain the asymptotic (overdamped) regime, governed by
transport equations, namely: for the particle's density, a
Smoluchowski-reactive like equation; for the particle's momentum density, a
generalized Ohm's like equation; and for the particle's energy density, a
Maxwell-Cattaneo like equation. Defining a nonequilibrium temperature as the
mean kinetic energy density, and introducing Boltzmann's entropy density via
the one particle distribution function, we present a complete
thermohydrodynamical picture for a charged Brownian gas. We probe the validity
of the local equilibrium approximation, Onsager relations, variational
principles associated to the entropy production, and apply our results to:
carrier transport in semiconductors, hot carriers and Brownian motors. Finally,
we outline a method to incorporate non linear reactive kinetics and a mean
field approach to interacting Brownian particles.
