文档介绍：J. Phys. A: Math., Nucl. Gen., Vol. 7, No. 15, 1974. Printed in Great Britain. 0 1974
Non-equilibrium statistical mechanics I.
The Boltzmann transport equation
John M Blatt and Alex H Opie
Applied Mathematics Department, University of New South Wales, Kensington, NSW,
Australia
Received 28 March 1974
Abstract. By using a new method, the Boltzmann transport equation is deduced from the
Liouville equation. The method involves an assumption of a statistical nature, which can
be relaxed so as to yield equations more general than the Boltzmann equation, or than
presently known extensions of the Boltzmann equation. The method also clarifies the
origin of the time irreversibility of the Boltzmann equation.
1. Introduction
Approximately one hundred years ago, Ludwig Boltzmann set up his transport equation
for dilute gases (Boltzmann 1872, 1875) by using intuitive arguments. The Boltzmann
equation is the basis for nearly all discussions of transport properties in dilute gases,
and there is excellent agreement between its prediction and experiment.
heless, problems remain even one hundred years later ; the most important
ones are:
(i) The Boltzmann equation leads to time irreversibility, ie, the forward sense of
time is distinguished, and memory of the initial state of the system at time t = 0 is lost
as the solution proceeds towards thermal equilibrium. By contrast, the equations of
classical dynamics are reversible in time, and the question arises how one can derive
an equation with irreversible results (Boltzmann’s equation) from a time-reversible
basic equation (Liouville’s equation for the full distribution function). In Boltzmann’s
own derivation, the irreversibility can be traced back to a statistical assumption about
the probability of collisions (the ‘Stosszahlansatz’); but it would be more satisfactory
to have the statistical assumption appear directly related to quantities involved in the
Liouville