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Momentum Balance


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




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.

next up previous
Next: Detailed Balance Up: DESIGN AND ANALYSIS Previous: Continuity Equation

William R. Frensley
Thu Jun 8 17:53:37 CDT 1995