A generally accepted approach to the problems of statistical physics is to begin with the general theory of many-body dynamics and to proceed by deductive reasoning to a formulation which provides an answer for the problem of interest (see, for example, Reichl, 1980). The steps in this deductive chain necessarily involve the introduction of extra assumptions in the form of suitable approximations. One may loosely categorize the levels of approximation in terms of the independent variables which are required to specify the state of a system. The most detailed level is the fundamental many-body theory, which in principle requires a complete set of dynamical variables for each particle. This can be reduced to the kinetic level by restricting one's attention to one- or two-body properties [by truncating the BBGKY hierarchy of equations, for example (Reichl, 1980, sec. 7C)]. It may also be necessary to remove from explicit consideration other dynamical variables of the complete system, such as photon or phonon coordinates, when electrons are the particles of interest. The kinetic theory is expressed in terms of distribution functions defined on a single-particle phase space, requiring one position and one momentum variable for each spatial dimension. (In the quantum case, this goes over to two arguments of the density operator.) The hydrodynamic level of approximation is obtained by making some assumption about the form of the distribution function with respect to momentum, and integrating over all momenta. Thus, the hydrodynamic theory is expressed in terms of densities which are functions of position only.
The approach taken in the present work is quite different from the conventional deductive approach. The objective is to identify the mathematical properties which are required of simple kinetic models of open systems. The procedure will be to construct small, spatially discretized models and to numerically explore their properties. The significance of the results must then be argued inductively.