Kinetic theory appears to be the simplest level at which one may
consistently describe both quantum interference and irreversible
phenomena (Prigogine, 1980). The only available levels which are simpler,
in that they require fewer independent variables, are
hydrodynamics and elementary (single-particle, pure-state) quantum
mechanics. Hydrodynamics (as embodied in Ohm's law and the
drift-diffusion equation in solid-state physics) provides no means to
describe quantum effects such as resonance phenomena because it retains
no information on the distribution of particles with respect to energy
or momentum.
On the other hand, if one attempts to include irreversible processes
within the framework of elementary quantum mechanics, the continuity
equation is most often violated. Irreversible processes will generally
result in the time dependence of some physical observable showing an
exponential decay. The only time dependence provided by elementary
quantum theory is the dependence of the wavefunction.
Exponential decay implies that **E** must have a negative imaginary part,
which means that the electron (for example) exponentially disappears,
violating charge conservation. As we have seen, violations of
continuity still occur when the irreversible processes are described by
the Fermi golden rule or Pauli master equation (see Appendix
10). To maintain consistency with the continuity
equation, we must allow off-diagonal elements of the density matrix (in
the eigenbasis of the Hamiltonian) to develop as the system evolves (see
Peierls, 1974).
Because we do not know * a priori* which off-diagonal elements are
required, we must admit all off-diagonal elements. A theory which
describes the evolution of the complete (single-particle) density
operator, including the off-diagonal elements, is by definition a
kinetic theory.

To express this point in another way, * we cannot, in general,
assume that the particles in an irreversible system occupy the
eigenstates of the Hamiltonian.* The proper basis states for a
one-particle description are the eigenstates of the density operator,
and thus the specification of the basis set should be a result obtained
from a proper theory, rather than an * a priori* assumption in the
theory. The exception to this situation is the particular case of
thermal equilibrium. In this case we know that the density operator is
a function of the Hamiltonian (via the Bloch equation, ), and if an
effective one-particle Hamiltonian is an adequate description, the
particles in the system will be found in eigenstates of this
Hamiltonian, if they are in equilibrium.

The usual way to describe the effects of irreversible or dissipative
processes at the kinetic level is to add a
collision term (of one form or another) to the Liouville equation
(2.3) to obtain a Boltzmann equation. This is a valid
procedure so long as the dissipative processes are sufficiently weak
that the motion of the particles can be viewed as periods of free flight
interrupted by collision events. Such a term takes its
simplest form for interactions between the particles of
interest (* i.e.*, electrons) with particles which are either
spatially fixed (such as impurities in solids) or which can be modeled
as components of a thermal reservoir (such as the phonons). In this
case (and within the Markov assumption) the collision term is a simple
linear superoperator expression and we can write the Boltzmann equation
as

where is the collision superoperator. (We will see later what condition must satisfy to preserve the continuity equation.) For two-body collisions the operator is a more complex object, operating on a two-body density matrix or (if the Stosszahlansatz is invoked) a product of one-body density matrices which introduces nonlinearity.

A characteristic feature of irreversible systems is the existence of stable stationary states, which can be either the equilibrium state or a nonequilibrium steady state if the system is driven by an external agency. Perturbations upon such a steady state will, in general, decay. To describe this decay the Boltzmann superoperator must have eigenvalues with negative real parts. In the usually studied case the Liouville superoperator is Hermitian, so by itself would produce purely imaginary eigenvalues. The collision operator introduces the negative real parts of the eigenvalues. Physically, we expect that there should be no eigenvalues with positive real parts, because these would correspond to exponentially growing modes, and the system would not be stable. The presence of eigenvalues with negative real parts together with the absence of eigenvalues with positive real parts implies that the system is time-irreversible.

The study of the fundamental origins of irreversibility in physical theory remains an area of active discussion and debate, more than a century after the question was first raised. However, if one's objective is to develop useful models of physical systems with many dynamical variables, rather than to rigorously construct a deductive mathematical system, it is clearly most profitable to adopt the view that irreversibility is a fundamental law of nature. For the present purposes a more precise statement of this law is that ``simple'' systems will always stably approach a steady state. In this context simple systems are those which can be regarded as being composed of a single type of particle or single chemical species and such that all other types of particle or excitation can be represented by thermal reservoirs. [Multicomponent systems can display exponential growth or stable oscillation (Prigogine, 1980).] The stability of the physical system implies that the kinetic superoperator which generates the time evolution of the density matrix (whether it is of the Liouville, Boltzmann, or some other form) cannot posses eigenvalues which would lead to growing exponential solutions. That is, there can be no eigenvalues with a positive real part. This condition will determine the sort of boundary conditions that can be used to model open systems.

Throughout most of the present analysis the collision terms will be neglected, because we will see that irreversibility enters through the open-system boundary conditions. The irreversible open-system model permits a wide variety of phenomena to be described at least qualitatively without invoking a collision term. This is not to say that irreversible collisions or dissipative interactions within a system are not significant effects. Indeed, a central thrust of traditional transport theory is the derivation of kinetic descriptions of such phenomena. The present neglect of the collision term is merely for the sake of simplicity, and it should be borne in mind that such a term may be readily added to any of the calculations to be discussed (see Appendix 14).

Thu Jun 8 17:53:37 CDT 1995