I have remarked that the Markovian kinetic models considered here are not equivalent to the usual elementary quantum-mechanical models of systems such as tunneling diodes. Let us now explore the differences between these two types of models by examining how they may be viewed as different approximations to a single many-body theory. In the approach to many-body transport theory developed by Kadanoff and Baym (1962) and by Keldysh (1964) and elaborated by Langreth (1976) and by Mahan (1987) the description of a quantum system is contained in a Green function,
where is the field operator. The density operator can be obtained from
Note, however, that the Green function has, in general, a second time argument , and this supplies the additional degree of freedom required to describe non-Markovian behavior. The demonstration of the correspondence between the Green function formalism and more classical transport equations proceeds applying a Wigner-Weyl-like transformation to the time variables: Define new variables and , and then Fourier transform with respect to :
In the absence of interactions the equations of motion for then become (Mahan, 1987), in the present notation,
[If interactions are present, collision terms involving the self-energy appear on the right-hand sides of (6.108) and (6.108).] Without interactions (6.108) is just the Liouville equation and (6.108) is a symmetrized Schrödinger equation. On an unbounded domain, these equations simply reproduce pure-state quantum mechanics, as noted above, and the usual tunneling theory follows. However, if we restrict the domain so as to obtain the open-system case, and we wish to reproduce the tunneling theory, we would have to apply traveling-wave boundary conditions such as those discussed in Appendix 12. Such boundary conditions necessarily introduce a dependence upon into (6.108). Even though we are still considering a ``noninteracting'' system (in the usual sense of no dissipation), we see that additional -dependent boundary terms must appear in (6.108) and (6.108).
The Markovian models neglect this -dependence. They are thus not equivalent to the tunneling or scattering theory. One can view such models either as an approximation to the tunneling theory, or alternatively, as simply a different approximation to the underlying many-body theory. In the latter view, the steady-state tunneling theory is obtained by neglecting the T-dependence of , whereas the Markovian model is obtained by neglecting the -dependence of . Thus, we may regard the Markov approximation as an a priori assumption that is independent of . Inverting the Fourier transform (6.106) shows that this is equivalent to assuming
This makes explicit the Markov assumption that the evolution of the system does not depend upon its past history.
To establish the plausibility of the Markov assumption (6.109), let us again consider the picture of an open system as a finite segment of length l of a much larger ``universe'' of length L which is occupied by a free electron gas. The Green function for this non-interacting system is
where is the probability that state k is occupied and is the energy of that state. Now, by examining within the system itself (that is, over and ) we cannot resolve the wavevectors of any excitations to an accuracy better than . On the other hand, because the ``universe'' is of a much larger length L, there will actually be many wavevector states within any such interval. Thus, the that one would observe within the system will be an average over these states of the form:
Now, using , where is the velocity of state k, we can change the integration variable to an energy, and perform the integral to obtain:
The bracketed factor approaches as . Now, l is fixed, of course, and thus the width of the ``-function'' is fixed. Moreover, the width is just the transit time across the system at the given k. This suggests the interpretation of (6.112): Any excitation within the system will propagate away (out of the system), and thus its temporal correlation function will decay after a time of the order of the transit time across the system. This demonstrates the motivation for the Markov assumption (6.109) and also its limitation. The generalization of the present open-system model beyond the Markov approximation has not yet been attempted and would be an obvious task for the further development of this approach. [The initial steps in this direction might be found in the work of Ringhofer, Ferry, and Kluksdahl (1989), who study the formulation of non-reflecting boundary conditions for the Wigner function. This work, however, is concerned primarily with obtaining local (in space and time) approximations to the rigorously nonlocal problem.]