The present work employs numerical computation and modeling for a purpose for which it is not often employed: as the primary mode of investigating the structure and consequences of a physical theory. The more traditional mode of investigation is, of course, to maximize the use of analytical mathematics and resort to numerical techniques only when the opportunities for analysis are exhausted, or when it is necessary to evaluate those complicated expressions which express an analytical solution. Any particular approach to describing physical phenomena will be successful only for some subset of those phenomena and will be otherwise ineffective. Because analytical mathematics is such a widely used tool, its domain of success has been extensively explored; this domain consists of those problems with sufficient symmetry to admit analytic solutions and those problems which can be regarded as small perturbations on analytically soluble problems. For statistical phenomena this generally means thermal equilibrium of analytically tractable systems and very small departures from equilibrium. Numerical simulation techniques which are inherently nonperturbative are better able to address more complex structures and/or far-from-equilibrium states. Because the study of discrete numerical models is not widely practiced, it is worth examining the principles by which such models may be constructed, using the present open-system model as an example.
A common point of view is to regard discrete numerical models, such as finite-difference models for partial differential equations, as approximations to the ``truth'' embodied in the continuum formulation of the problem (for example, Lapidus and Pinder, 1982). Such a discrete model can represent the continuum solution only to within an accuracy which is proportional to some power of the mesh spacing (or other appropriate measure of the coarseness of the discrete model). This tends to lead one to believe that the physics of the situation can be represented only to a given order of accuracy, so that such expressions as conservation laws (or balance equations) will be satisfied only to that order (see, for example, Aubert, Vaissiere, and Nougier, 1984). A corollary to this view is that higher-order approximations produce better models. Such is often not the case (Press et al., 1986), because higher-order approximations usually admit spurious short-wavelength modes which adversely affect both the stability and accuracy of such models.
In fact, a better guiding principle is to seek discrete models which are constructed so as to exactly satisfy the physical laws which govern the behavior of the real system. In practice, one often finds that it is only possible to satisfy some, but not all of these laws. Which laws are exactly satisfied and the order of the error terms in the remaining laws depend upon the details of the particular discretization scheme. This situation has led to the conventional wisdom that the discretization of partial differential equations is ``an art as much as a science'' (Press et al., 1986). The science which is often lacking is a consistent analysis of the degree to which all reasonable discretization schemes satisfy the appropriate laws, or preferably the identification of one scheme which exactly satisfies the relevant laws. A particularly attractive example of the latter situation has been given by Visscher (1988, 1989). It is a discretization of Maxwell's equations in three dimensions which exactly satisfies the integral forms of the equations. This is accomplished by assigning the various field quantities (charge and current density, electric and magnetic field) appropriately to the centers, faces and edges of cubic finite-difference cells. Unfortunately, we shall see that this ideal situation is not likely to apply to kinetic open-system models, and some trade-offs must be made between the different laws that we wish to satisfy.
A systematic way to determine the advantages and limitations of a discrete model is to first identify the physical laws which the model ought to satisfy and then evaluate the order of the errors by which the discrete model fails to satisfy those laws. For the present open-system model, I assert that there are four such laws: (i) charge continuity, (ii) momentum balance, (iii) detailed balance of the equilibrium state, and (iv) stability of nonequilibrium states. Energy balance is not included in this list because it adds no physics that is not already described by momentum balance so long as we neglect energy-redistributing processes such as electron-electron or electron-phonon scattering. Condition (iv) is just the criterion that we have examined extensively, that none of the eigenvalues of the Liouville operator should have a positive imaginary part.