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.]

Thu Jun 8 17:53:37 CDT 1995