Studies in the C*-Algebraic Theory of Nonequilibrium Statistical Mechanics: Dynamics of Open and of Mechanically Driven Systems

Abstract

We construct a C*‐algebraic formulation of the dynamics of a pair of mutually interacting quantum‐mechanical systems S̃ and Ŝ, the former being finite and the latter infinite. Our basic assumptions are that: (i) Ŝ, when isolated, satisfies the Dubin‐Sewell dynamical axioms; (ii) the coupling between S̃ and Ŝ is energetically bounded and spatially localized; and (iii) the initial states of S̃ and Ŝ are mutually uncorrelated, with S̃ in an arbitrary normal state and Ŝ in a Gibbs state. Our formulation leads to a rigorous theory of (a) the dynamics of a finite open system, i.e., of a finite system (S̃) coupled to an ``infinite reservoir'' (Ŝ), and of (b) the dynamics of an infinite system (Ŝ), driven from equilibrium by a ``signal generator'' (S̃). As regards (a), we show that the state of S̃ always remains normal, and we derive a generalized master equation (in an appropriate Banach space) governing its temporal evolution. As regards (b), we show that the state of Ŝ always corresponds to a (time‐dependent) density matrix in the representation space of the algebra of observables for Ŝ, induced by the initial Gibbs state. By formulating the linear part (appropriately defined) of the response of Ŝ to S̃, we generalize the fluctuation‐dissipation theorem to infinite systems. Further, we show that the total effect of S̃ on Ŝ reduces to that of a ``classical'' time‐dependent external force in cases where the initial state of S̃ possesses certain coherence properties similar to those of the Glauber type.