Thermodynamic probability theory: Some aspects of large deviations

John Lewis, DIAS Dublin
Thursday 22 and Friday 23 October 1998, Cardiff University

  1. Thursday 2.00 pm : The entropy analogy N/1.32 Trevithick

  2. Friday 1.30 pm : Large deviation theory S/1.28 Trevithick

  3. Friday 3.00 pm: Some applications S/1.28 Trevithick

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.

  1. Review article
    J.T. Lewis, C.E. Pfister
    ``Thermodynamic probability theory: Some aspects of large deviations''. Russian Math. Surveys 50: 2, 279-317 (1995)

  2. 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, 319-386 (1995)

  3. 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, 1935-1947 (1997)

  4. Applications to telecommunications

  5. 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, 37-65 (1993)

    D.E. Evans MATHS
    J. Inglesfield PHYSX
    University of Wales, Cardiff