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.

Thu Jun 8 17:53:37 CDT 1995