Thermodynamic probability theory:
Some aspects of large deviations
John Lewis, DIAS Dublin
Thursday 22 and Friday 23 October 1998, Cardiff University
 Thursday 2.00 pm : The entropy analogy N/1.32 Trevithick
 Friday 1.30 pm : Large deviation theory S/1.28 Trevithick
 Friday 3.00 pm: Some applications S/1.28 Trevithick
Abstract:
Boltzmann's remarkable formula S = k ln W, relating the
thermodynamic entropy S of a macroscopic equilibrium state to the
number W of microscopic states corresponding to the macroscopic
state, caused problems for theoretical physicists: the entropy is a
function of a small number of variables such as u, the internal
energy per unit volume, while W must be interpreted as a measure on
phase space. To have any hope of giving the formula a precise
meaning, we should pass to the the limit in which the volume of the
system
becomes infinite and hope to get a formula for s(u), the entropy per
unit volume. The problem was resolved in 1965, when Ruelle gave a
good
definition of s(u) and proved the existence of the relevant limit
for
a large class of systems. The function derived in this way has all
the
properties required of an entropy function in Gibbs axiomatization
of
equilibrium thermodynamics. These ideas were developed by Ruelle and
Lanford to provide a rigorous treatment of statistical
thermodynamics,
described in detail in Lanford's 1971 Batelle Lectures.
Ruelle's idea
turned out to have a surprising ramification in probability theory,
leading to important developments in the theory of large deviations.
Large deviation theory began as an attempt by Khinchine to refine the
central limit theorem. However, the theory we shall describe starts
with Cramer's refinement of the week law of large numbers. In a
digression in his lectures on statistical thermodynamics, Lanford gave
a completely new proof of Cramer's theorem. This idea was quickly taken
up by probabilists, leading to powerful generalizations of Cramer's
theorem.
By now, large deviation techniques are used widely for the rigorous
treatment of problems in mathematical physics. This development came
from a different direction: in the early 'sixties, Donsker proposed
using such a refinement of the weak law of large numbers to give a
generalization of Laplacian asymptotics to function spaces; this was
given a precise form in the definitive work of Varadhan. Large
deviation theory has applications in other areas; it illuminates
classical results in information theory and ergodic theory and its use
in queueing theory leads to very practical applications in problems of
resource allocation in broadband telecommunication networks.
The aim of these lectures is to illustrate the interplay of
thermodynamic and probabilistic ideas in large deviation theory. The
first lecture will attempt to survey this interplay of ideas without
going into any technical details and consequently will be addressed
to a wider audience than will the second and third lectures. The
second lecture will explain the main ideas of large deviation theory
and its applications to information theory and ergodic theory. The
third lecture will describe applications of large deviation theory to
the problem of equivalence of ensembles in statistical mechanics and
to algorithms for connection admission control in broadband
telecommunication networks.
 Review article
J.T. Lewis, C.E. Pfister
``Thermodynamic probability theory: Some aspects of large deviations''. Russian Math. Surveys 50: 2, 279317 (1995)
 Conditional limit theorems
J.T. Lewis, C.E. Pfister, W.G. Sullivan
``Entropy, concentration of probability and conditional limit theorems''. Markov Processes and Related Fields 1, 319386 (1995)
 Applications to information theory
J.T. Lewis, C.E. Pfister, R. Russell and W.G. Sullivan
``Reconstruction sequences and equipartion measures: An examination of the asymptotic equipartition property''. IEEE Transactions on Information Theory, vol. 43, No. 6, 19351947 (1997)

Applications to telecommunications
 Applications to quantum statistical mechanics
T.C. Dorlas, J.T. Lewis and J.V. Pule
``The full diagonal model of a Bose Gas''. Commun. Math. Phys. 156, 3765 (1993)
D.E. Evans MATHS
J. Inglesfield PHYSX
University of Wales, Cardiff