** Next:** INTRODUCTION

This hyperdocument was derived from an article published in Reviews of Modern Physics,
Vol. 62, pp. 745-791 [1990].

# Boundary Conditions for Open Quantum Systems Driven Far from
Equilibrium

**William R. Frensley **

### Abstract:

*This is a study of simple kinetic models of open
systems, in the sense of systems which can exchange conserved particles
with their environment. The system is assumed to be one-dimensional and
situated between two particle reservoirs. Such a system is readily
driven far from equilibrium if the chemical potentials of the reservoirs
differ appreciably. The openness of the system modifies the
spatial boundary conditions on the single-particle Liouville-von Neumann
equation, leading to a non-Hermitian Liouville operator. If the
open-system boundary conditions are time-reversible, exponentially
growing (unphysical) solutions are introduced into the time-dependence
of the density matrix. This problem is avoided by applying
time-irreversible boundary conditions to the Wigner distribution
function. These boundary conditions model the external environment as
ideal particle reservoirs with properties analogous to those of a
blackbody. This time-irreversible model may be numerically evaluated in a
discrete approximation, and has been applied to the study of a
resonant-tunneling semiconductor diode. The physical and
mathematical properties of
the irreversible kinetic model, in both its discrete and continuum
formulations are examined in detail. The model demonstrates the
distinction in kinetic theory between commutator superoperators, which
may become non-Hermitian to describe irreversible behavior, and
anticommutator superoperators, which remain Hermitian and are used to
evaluate physical observables.
**
*

** CONTENTS**

*William R. Frensley *

Thu Jun 8 17:53:37 CDT 1995