To begin the analysis of the irreversible open-system model defined by (4.51), let us consider the continuity equation (6.95). First define the discrete approximation to the local particle density in the obvious way, converting the integral in (6.92) to a sum:

In a discrete model the current density is most naturally regarded as a
quantity which is defined on each * interval* between adjacent mesh
points, rather than on the mesh points themselves. Thus, the divergence
of the current density is a difference taken between adjacent intervals
and is associated with their common mesh point. Let us denote the
current on the interval between and by .
Then if **J** is to exactly satisfy a discrete continuity equation
we must define to be

The moment of the Liouville equation becomes

To show that the contribution from the potential operator vanishes,
let us consider the sum over **k** first. The sum can be reordered and
then can be expanded using equation (4.43):

Now, this sum will vanish if

which happens if , and was defined so as to satisfy this relation. This is the Fourier completeness relation mentioned earlier. Thus, the discrete model exactly satisfies the continuity equation

The only limit on the precision of this relationship is the arithmetic roundoff error, which is generally several orders of magnitude smaller than typical discretization errors.

Satisfying the continuity equation via the Fourier completeness
relation (7.117) relies upon the special properties of the
(artificial) Brillouin zone created by the **q**-discretization.
To see this, consider **k** and such that
. The term should describe the effect of
a short-wavelength component, but because of the ambiguity introduced by
the discretization the term is really derived from the much
longer-wavelength component indexed by . Such an
effect is called ``aliasing'' in the context of signal processing and
sampling theory (Oppenheim and Schafer, 1975, sec. 1.7), where
it is generally regarded as undesirable, and it is mathematically the
same as an ``umklapp
process'' in the context of solid-state physics.
The derivation of the continuity equation in the continuum case relies
on no such property; it follows directly from the antisymmetry of the
potential kernel . In a finite model, however, we must cut off
the sequence of **k**'s at some value, and this will remove some terms
which would need to be present in the summations of the second term of
(7.115) in order to make this term exactly vanish by antisymmetry.
Thus, if we do not rely upon the Fourier completeness property, the best
we can hope for is to satisfy the continuity equation to . The
error can be made numerically very small by proper choice of the
limiting values of **p**, but, formally, the continuity equation would not
be exactly satisfied.

Thu Jun 8 17:53:37 CDT 1995