 
Summary: DISSIPATIVE DYNAMICS IN SEMICONDUCTORS AT LOW
TEMPERATURE
GEORGE ANDROULAKIS, JEAN BELLISSARD, AND CHRISTIAN SADEL
Abstract. A mathematical model is introduced which describes the dissipation of elec
trons in lightly doped semiconductors. The dissipation operator is proved to be densely
defined and positive and to generate a Markov semigroup of operators. The spectrum
of the dissipation operator is studied and it is shown that zero is a simple eigenvalue,
which makes the equilibrium state unique. Also it is shown that there is a gap between
zero and the rest of its spectrum which makes the return to equilibrium exponentially
fast in time.
The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model
is meant a mathematical construct which, with the addition of certain verbal interpretations, describes
observed phenomena. The justification of such a mathematical construct is solely and precisely that it is
expected to work. (J. von Neumann [51])
1. Introduction
This article is dedicated to the construction and the fundamental properties of a model of
dissipative transport, describing the electron or hole transport in semiconductors at very
low temperature. By "very low" it is meant that the temperature is low enough so as
to confine the charge carriers to the impurity band. Without dissipation, the transport
is coherent and is likely to be described by an Anderson model, namely a Schršodinger
