The central conclusion of the present work is that an open system, in the sense of one which exchanges particles with its environment through spatially localizable interfaces, is necessarily irreversible. The reasoning behind this conclusion is a reductio ad absurdum argument. We have seen that a particular reversible model of an open system possesses unphysical instabilities. The mathematical properties underlying these instabilities, the existence of complex eigenvalues of non-Hermitian superoperators and the requirement that these occur in conjugate pairs due to time-reversal symmetry, are sufficiently general that we should expect such instabilities in any reversible model. Thus, physically acceptable models of open systems must be inherently time-irreversible.
A particular class of irreversible open-system models was presented, and the stability of the resulting solutions was demonstrated. The irreversibility of these models follows from making a distinction between particles entering and leaving the system. Similar ideas, generally applied in the time domain, are the basis for the established theories of irreversibility and dissipation. The present work demonstrates that spatial boundary conditions can be used to introduce irreversibility in a way which is very similar to that by which temporal initial conditions do so.
The present study of the kinetic theory of open systems helps to clarify the roles of superoperators generated by the commutator and anticommutator of a physical observable. It was demonstrated that, at the kinetic level, only the commutator superoperators should acquire non-Hermitian parts to model irreversible phenomena. Anticommutator superoperators remain Hermitian and are used to evaluate expectation values.
Some of the more mathematical issues concerning the properties of the present open-system models remain unresolved, particularly the question of positivity of the resulting Wigner distribution functions. However, the results obtained by applying these models to the resonant-tunneling diode demonstrate the usefulness and credibility of this approach.
This work is certainly not an exhaustive examination of the theory of open systems. Undoubtedly, many more approaches to the subject can be formulated. However, one should note that the significant behaviors of an open system involve a strong coupling between the system and its environment and large deviations from equilibrium within the system. It thus appears unlikely that perturbative approaches will contribute much to the theory of such systems. Other analytic approaches will be effective only in cases displaying some exceptional symmetry (and of course the present definition of open system rules out translational symmetry). It thus appears that numerical models such as those examined here will probably be the mainstay of such investigations.