One begins to encounter the limits of a simple discrete model when the momentum balance (first moment) equation (6.96) is considered. To evaluate the rate of change of current density, insert the discrete Liouville equation (4.51) into the definition of (7.114). One then obtains

where

Note that the requirements of consistency in the discretization scheme
imply that , which one might expect to depend only upon the
values of **f** at , actually depends upon the values of **f** at
and . This sort of spreading over the domain becomes worse as
higher moments are considered. It is probably more correct to regard
as a local function of **q** and attribute an error of to
(7.119) (because ).
Now consider the potential terms in
(7.119). For simplicity, let us neglect the
different **j** indices required by the form of **J** and simply evaluate

where equation (4.43) is again used and the sums reordered as
before. Now, in the continuum case (6.96) this expression
reduces to . The discrete expression
(7.121) shows a functional of (the first bracketed factor)
times **n**. If we consider only the first term of the sum over
and take for small , we get
, which is just the centered-difference
approximation to . However, the other terms of
the sum are not negligible. While is small, the higher
terms just add in more remote approximations to .
Of course, approaches zero much more rapidly than
as approaches . Thus, there is a natural cutoff of these
higher terms so long as .
This helps
to explain the significance of the limit of the summation of
(4.43). The value of was originally chosen for the
upper limit of this sum on the purely empirical basis that the results
were most credible with this value, and multiples of were
investigated because the summation is carried out in position space.
However, most calculations have taken , so these
conditions are approximately equivalent. The significant
result is that the momentum balance equation (6.96) is not
satisfied exactly by the discrete model.

The conformance of the discrete model to the momentum balance equation can be significantly improved by modifying the form of the discrete potential operator (4.43). However, this must be approached with some care. One could, for example, simply discretize the classical form and if this is done properly, momentum balance will be exactly satisfied. The problem with this approach, of course, is that it discards any quantum interference effects. Mains and Haddad (1988a, 1988c) have suggested a better approach. They recommend an alternative expression for which leads to a model which exactly satisfies a discrete momentum balance expression. The idea is to weight the expression for as

If we now evaluate the first moment of the potential term we find

exactly. The use of a weighting function in momentum space corresponds
to a convolution in position space. If we reinterpret (7.122) as a
continuum expression, a bit of manipulation will show that (7.122)
can be derived from a ``smoothed'' potential , where the convolution function **w** is just a
rectangular pulse on the interval . It can be written as
, where denotes
the Heaviside step function. Qualitatively, the effect of this scheme is to
smooth out any abrupt change in the potential so that any such change is
distributed over at least two mesh intervals. However, the convolution
theorem does not hold exactly in the finite, discrete domain of the
present problem. One consequence of this is that the discretization
based upon (7.122) does not exactly satisfy the continuity equation
via the Fourier completeness relation (7.117), but does so only to
, as discussed above.

A related idea is to use some form of ``data windowing'' (Oppenheim
and Schafer, 1975, sec. 11.4) in the
evaluation of the discrete potential superoperator. This technique is
used in the Fourier analysis of finite sets of sampled data, and
in the present context would involve multiplying the
factor in (4.43) by some
function of which decreases to zero for large (the window
function). That is, the weighting would be done in real space rather
than in **k** space. The objective of data windowing is to maximize the
fidelity of the Fourier spectra derived from a finite set of data to
those of a hypothetical infinite data set by minimizing the spurious
effects associated with cutting off the data at some finite value.
Qualitatively, this would seem to suit the requirements of discrete
models of quantum systems. Invoking the idea that
encodes the quantum-interference effects, we might also
interpret a data windowing procedure as an approximate description of
the continuous loss of coherence as one examines points separated
farther apart in a dissipative system. This procedure might provide a
way to interpolate between the quantum and classical regimes, whereas
the obvious schemes for doing so with the Wigner function, expanding in
powers of , are known to fail (Heller, 1976). These are intriguing
possibilities, but the effects of data windowing on
the present sort of open-system models have not been extensively
investigated.

Thu Jun 8 17:53:37 CDT 1995