Soluble idealised models in particle transport theory

Abstract

The central aim of Part I of this thesis is to investigate non-equlibrium processes in physics by studying the so called Rayleigh's Piston model which was originally conceived by Lord Rayleigh in 1891. In its most general aspect the modern formulation involved the study of the 'Master Equation' for the statistical evolution of an ensemble of test-particles (mass M) constrained to move in one dimension interacting with heat-bath particles (mass m). By using the numerical techniques developed in studying neutron thermalisation, we have investigated the accuracy of Rayleigh's original treatment or the so called Brownian limit and obtained numerical results for velocity autocorrelation function S_V(t) and electrical conductivity cr(o).

It is in the case of special Rayleigh's model where the masses are equal (M=m) that we have been able to solve the model exactly both by using the method of singular eigen functions and by the method of Laplace transform. Thus a definitive connection is made with methods developed in the 'Linear Transport Theory' to solve problems in field of radiative transfer, neutron diffusion, the theory of plasma as well as elsewhere. For the special model, we have investigated the 'Velocity' barrier problem, the spatial problem and obtained exact expressions for the autocorrelation function, the diffusion constant, the electrical conductivity by using the linear response theory and tested the validity of the so called 'Gaussian Approximation' by examining moments of the Van Hove correlation function G(r, t).

In Part II of this thesis we have investigated the behaviour of a model consisting of an ideal charged electron gas in a uniform magnetic field and confined by a cylinderically symmetric potential. We have obtained exact expressions for the current density, the magnetic moment, the magnetic susceptibility and examined in detail the boundary effects.