In the kinetic level of description of a complex system, the effects of those degrees of freedom which are of less interest in a given problem are included implicitly in objects such as collision operators or effective interaction potentials. In the example of electronic devices such degrees of freedom should include electron coordinates outside the device, but within the external circuit. They also include all excitations of the device material apart from the single-electron states ( e.g., the phonons). Thus, at this level the state of the system is described by a one-body density operator or distribution function. In general, this can be written as
where i labels a complete set of states and the are real-valued probabilities for the system to be in state . Because we will be considering open systems in which the number of particles is not fixed, the usual convention for the normalization of ( and ) is not useful. Instead, we will adopt a normalization convention such that gives the actual particle density (in units of particles per , for example). More formally, is the one-body reduced density operator which is defined on a single-particle Hilbert space (Reichl, 1980, ch. 7). The complete density matrix defined on the many-particle Fock space (second quantization) may still be normalized to unity. The focus upon a single-particle description requires that one exercise some care concerning the quantum statistics. For example, if the equilibrium density operator is obtained by solving the Bloch equation, , the result will satisfy Maxwell-Boltzmann statistics. A similar calculation in the Fock space will, of course, satisfy Fermi-Dirac statistics.
For a system described by a simple single-particle Hamiltonian,
the time evolution of the density matrix is given by the Liouville-von Neumann equation:
where is the Liouville superoperator. The simplest approach to modeling the behavior of open systems is to apply the Liouville equation to a finite spatial domain representing the system of interest and to apply boundary conditions which model the openness of the system. The difficulties and ultimate success of this approach involve the effect that such boundary conditions have upon the properties (particularly the eigenvalue spectrum) of the Liouville superoperator.