  We investigate a kinetic model of a system in contact with several thermal
reservoirs at different temperatures $T_\alpha$. Our system is a spatially
uniform dilute gas whose internal dynamics is described by the nonlinear
Boltzmann equation with Maxwellian collisions. Similarly, the interaction with
reservoir $\alpha$ is represented by a Markovian process that has the
Maxwellian $M_{T_\alpha}$ as its stationary state. We prove existence and
uniqueness of a non-equilibrium steady state (NESS) and show exponential
convergence to this NESS in a metric on probability measures introduced into
the study of Maxwellian collisions by Gabetta, Toscani and Wenberg (GTW). This
shows that the GTW distance between the current velocity distribution to the
steady-state velocity distribution is a Lyapunov functional for the system. We
also derive expressions for the entropy production in the system plus the
reservoirs which is always positive.
