Broadly speaking, there are two ways to set up a transport problem: the Eulerian approach in which the coordinates are fixed in the reference frame of the observer; and the Lagrangian approach, in which the coordinates are fixed in the reference frame of the transported fluid. The present work focuses upon the Eulerian approach. However, a number of formulations of quantum transport theory are expressed in terms of Lagrangian variables. These include the center-of-mass approach of Lei and Ting (1985) and the quantum Langevin-equation approach of Hu and O'Connell (1987). The accelerated basis states studied by Krieger and Iafrate (1986) adapt the Lagrangian variables to pure-state quantum mechanics. It appears that none of these approaches has yet been applied to an open-system problem in the present sense, so there has been no analysis of the effects of boundary conditions within the Lagrangian approaches. Moreover, it is not at all clear that such approaches are well adapted to the description of tunneling, where there is no classical trajectory. (Although in this connection one should note the work of Jensen and Buot, 1989b, in which the trajectories in a resonant-tunneling diode were inferred from a solution for the Wigner function.)
In the classical case, however, much of the work dealing with open systems (and most of the work treating electron transport in nonuniform systems) has been cast in terms of the Lagrangian variables. This includes both deterministic approaches, such as that of Baranger and Wilkins (1987), and stochastic approaches, such as the widely used Monte Carlo technique (Jacoboni and Regiani, 1983; Castagné, 1985; Reggiani, 1985; Constant, 1985). If we consider the boundary conditions in such approaches, it becomes apparent that the ``inflowing'' boundary conditions (4.36) will occur quite naturally. In the approach of Baranger and Wilkins the Lagrangian variables define the mean trajectories of the particles, so one must specify the initial conditions on the trajectory, which is the value of the distribution function at the point where the trajectory enters the domain. Thus, the boundary conditions are completely equivalent to (4.36).
In the case of the Monte Carlo technique the boundary conditions are determined implicitly by the details of the algorithm used in the calculation, and such details are often omitted from the published reports. To understand the relationship between the algorithm and the boundary conditions, let us consider the algorithms described by Lebwohl and Price (1971) and Hockney and Eastwood (1981) (which is also described by Castagné, 1985). Any electron leaving the domain of the Lebwohl and Price calculation is immediately replaced by another electron entering randomly from either contact, with an initial momentum chosen from a thermal distribution. Thus, the number of electrons in the system is fixed (and the fact that this leads to simpler and more efficient programs is the motivation for the Lebwohl and Price approach). A distribution function evaluated with this algorithm will satisfy boundary conditions of the form (4.36), but the values of the boundary distributions will not necessarily remain fixed, as they depend upon the rate at which electrons impinge upon the contact. To view the problem another way, the same algorithm would be obtained from a model in which the system is assumed to be periodic, but which has a very strong scattering process located at that plane where the system closes upon itself. Thus, this approach really describes a closed system, and the fixed number of particles within the system is an indication that the system is actually closed. A truly open system results if the particles entering the domain are chosen by an independent stochastic process, and the resulting distribution function would then satisfy (4.36) with fixed boundary distributions. The algorithm described by Hockney and Eastwood (1981) is almost of this form, though the rate of particle injection is adjusted in response to the nearby density.
The discussion of Monte Carlo algorithms and boundary conditions brings out an important point: The number of particles in an open system necessarily fluctuates. While I have not addressed fluctuation phenomena in the present work, a more complete description should deal with such effects.