Irreversible gibbsian ensembles

In this paper we continue the investigation of a model for the description of irreversible processes which we had proposed in an earlier publication. This model permits the construction of Gibbs-type ensembles for open systems not in equilibrium. The internal dynamics of the system that is engaged in a nonequilibrium process is assumed to be described fully by its Hamiltonian. Its interaction with its surroundings, i.e. the driving reservoirs, is described in terms of impulsive interactions (collisions). The reservoirs themselves possess definite temperatures, are inexhaustible, and are free of internal gradients (i.e. they are temperature baths). The ensemble obeys an integro-differential equation in Γ-space, containing both the terms of the Liouville equation and a stochastic integral term that describes the collisions with the reservoirs. It is shown in this paper that, under very general assumptions, all distributions approach each other in the course of time. If there exists a stationary solution, it will be unique and will be approached asymptotically by every time-dependent solution. In general the stationary state does not represent thermodynamic equilibrium; the ensemble remains unchanged only because its surroundings maintain temperature gradients inside the thermodynamic system. Only if these surroundings are all at one temperature, i.e. if the system is in contact with but one reservoir, then the stationary state will correspond to the canonical distribution. As a result, the stochastic integral kernel that describes the effect of collisions with the reservoir will satisfy certain symmetry conditions. A detailed investigation of our micro-model shows that these conditions are indeed satisfied if the reservoir components are themselves in a canonical distribution prior to collision. In the presence of several reservoirs at slightly different temperatures, the Onsager reciprocal relations are satisfied by the stationary distribution. In our model the Onsager relations are thus obtained without an appeal to fluctuation theory, and without the assumption that detailed balancing holds for the elementary stochastic processes, i.e. for the interactions between system and reservoir. In the latter part of the paper, finally, we consider reservoirs that maintain thermodynamic potentials in addition to the temperature, including chemical potentials. It turns out that our principal results are unaffected by this generalization.