The more active, and thus the more interesting, products of technology are systems which operate far from thermal equilibrium. An examination of a few examples of such systems shows that they are generally open, in the sense that they exchange matter with their environment. The present work examines some schemes by which open quantum systems (which are beginning to become technologically important in the context of microelectronics) may be effectively described at a kinetic level.
In the context of the present work, an ``open system'' is one which can exchange locally conserved particles with its environment. Moreover, we wish to focus upon the far-from-equilibrium behavior of such a system, and thus the definition of open system will be further restricted to mean one which is coupled to at least two separate particle reservoirs, so that a nonequilibrium state may be created and maintained. To specify such a system we must regard it as occupying a finite region of space, and thus the exchange of particles must consist of a current flowing through that surface which is taken to be the boundary of the system. It does not appear that the statistical physics of such a situation has been the subject of a close examination. [The traditional use of the grand canonical ensemble to define the equilibrium state (Tolman, 1938, sec. 140) contemplates a system coupled to a single particle reservoir.] There is a large body of work on quantum systems which are coupled to a reservoir so as to permit an exchange of energy (see, for example, Chester, 1963; Louisell, 1973; Haken, 1975; Davies, 1976; and Oppenheim, Shuler, and Weiss, 1977), or are in purely thermal contact with two or more reservoirs (Lebowitz, 1959). Most of these analyses are directed more to the problem of damping (as seen in ohmic conduction) than to openness in the present sense. Much of the work in this area has been motivated by the development of optical technology (Louisell, 1973; Haken, 1975) in which the present distinction between openness and damping is unnecessary because the particles of interest are massless bosons. In a laser, for example, the degrees of freedom of greatest interest are the normal modes of the radiation field. A single theoretical model, the damped harmonic oscillator, is used to describe both the loss of energy (photons) to the gain medium within the cavity and the loss of photons to the output beam (Gordon, 1967; Scully and Lamb, 1967). The analogous processes in an electronic resistor (an open system in the present sense) are the scattering of an electron by a phonon within the resistive material and the escape of an electron from the resistive material into a more highly conductive contact. The present work will concentrate upon the consequences of the latter process. The difference between the system of massive fermions and the system of massless bosons is that the fermion system is constrained by a local continuity equation, whereas the boson system (within the usual models) is not so constrained.