STUDIES IN IRREVERSIBLE THERMODYNAMICS, V. STATISTICAL THERMODYNAMICS OF A SIMPLE STEADY-STATE MEMBRANE MODEL

Abstract

We are currently interested in the question of the possible existence of a general thermodynanics of steady-state systems, neither restricted to near-equilibrium nor to local equilibrium. For this taski, it is very helpful to have available exact properties for some very simple models. We consider such a model in this paper. It is hoped that the treatment of this model, and others to be discussed subsequently, will stimulate some interest in the general question mentioned above, even though the moclels may not themselves lead to an answer. Another purpose of the present paper is to point out the relation of this work to that of Jaynes1 and Shannon and Fienberg2 on generalized (nonequilibrium) partition functions in statistical mechanics. A large number of related models, and methods of finding some of their steadystate properties, were discussed in the two previous papers of this series.' 4 The Model.-The system whose steady-state properties we shall investigate here is a membrane between two infinite baths, A and B, both at temperature T. Each bath is itself an equilibrium system, but in general the baths are not in equilibrium with each other. The steady-state system (mrembrane) consists of B independent and equivalent units (or channels) for transport of a single component from bath A to bath B. A unit can exist in four different states (Fig. 1). Figure 2 shows the allowed transitions between states and the assigned transition probabilities or rate constants. Thus, for example, if a unit is in state 0, the probability that it will undergo a transition to state 2 in a time interval dt is aBdt, where aB is a constant. The constants f (desorption) and K (hopping) are intrinsic properties of the system itself, but the a's (adsorption) also involve the baths: aA = IXA and aB =3 XB, where XA = e/A/kT (in bath A), etc. If we choose aA = aB, the steady state is an equilibrium state. But ordinarily we take aA 7? aB. We digress to remark that in a much more general treatment, the unit might be very large, even macroscopic, so that a unit becomes a "system" and a system becomes an "ensemble" (B -* 03). The states referred to above would then be energy eigenstates available to each "system" (as in the Pauli-van Hove master equation)i. The model outlined above is nodel 2 in pa-rt ITT,' with n = 2, except that, for simplicity, we shall here take the partition function for a moleecule aclsorbed on a site as q = 1 (ground state only; eniergy = 0). .Properties of the Mlodel. A stochastic argument leads immediately to