Mathematics Department
Statistical Mechanics and Quantum Probability Group

Research Activities

Many-particle classical and quantum systems, or small systems interacting with such large systems, are studied using the methods of equilibrium statistical mechanics (David Lavis), scaling limits (Larry Landau), and the new methods of statistical dynamics (Ray Streater). In addition to standard physical systems, many-neuron systems and brain function are also studied (Ton Coolen).

A general method for constructing non-linear dynamical systems obeying the first and second laws of thermodynamics is developed in Statistical Dynamics, by Ray Streater (Imperial College Press, 1995). An essentialrole is played by the non-linear irreversible mapping Q which is a generalization of the Boltzmann Stosszahlansatz. A number of research projects are available in the areas of non-linear heat equations, complicated chemically reacting systems, and reaction-diffusion equations for dense fluids.

The standard methods of equilibrium statistical mechanics are used to study phase transitions in lattice spin models, the computer package Maple being used to reduce the transfer matrix to block diagonal form. These methods are described in Statistical Mechanics of Lattice Systems, Volume I(2nd edition) and Volume II, by D.A.Lavis and G.Bell (Springer, 1998).

Large time or space-time scaling limits of quantum systems yield irreversible and classical behavior: On the Weak Coupling Limit for a Fermi Gas in a Random Potential, Reviews in Mathematical Physics 5 (1993) and Macroscopic Observation of a Quantum Particle in a Slowly Varying Potential, Annals of Physics 246 (1996), by Larry Landau. The time-evolution of a free quantum particle on a lattice is described by Bessel functions and a detailed study has yielded new monotonicity properties and precise bounds: Bessel Functions: Monotonicity and Bounds, to be published.

Staff:

Ton Coolen
Larry Landau
David Lavis
Ray Streater

Postgraduate Study

Recent Departmental preprints in statistical mechanics and quantum probability:
2002, 2001, 2000, 1999, 1998, 1997.


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