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.
