Stationary Nonequilibrium Gibbsian Ensembles

The general theory of a Gibbs ensemble representing a system in contact with its surroundings is applied to several concrete situations of interest. By an appropriate choice of heat reservoirs a simply modified Liouville equation is found to describe a heat conducting system. The stationary nonequlibrium -space ensembles which describe such a system are found explicitly for some cases. In the simplest cases these ensembles turn out to be canonical with a temperature that is a weighted average of the reservoir temperatures. For other systems, such as Brownian particles inside a fluid whose temperature is not uniform, we find the stationary ensemble to terms linear in the temperature gradient. From this we are led to discuss ensembles that will approximately represent an arbitrary heat conducting fluid. A more general proof than previously given for the asymptotic approach of the -space distribution to its stationary value is also presented.