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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).

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William R. Frensley
Thu Jun 8 17:53:37 CDT 1995