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.