Table of contents

This paper reviews the state of affairs in a modern branch of mathematical physics called probabilistic topology. In particular we consider the following problems: (i) we stimate the probability of trivial knot formation on a lattice using the Kauffman algebraic invariants and show the connection of this problem with the thermodynamic properties of 2D disordered Potts model; (ii) we investigate the limiting behavior of random walks in multiconnected spaces and on non-commutative groups related to knot theory. We discuss the application of the above-mentioned problems in the statistical physics of polymer chains. On the basis of non-commutative probability theory we derive some new results in the statistical physics of entangled polymer chains which unite rigorous mathematical facts with intuitive physical arguments.

The relation between the non-equilibrium thermodynamics and kinetic theory of rarefied gases is discussed. The phenomenological equations of generalized non-equilibrium thermodynamics are formulated using the fundamental relations of kinetic theory. Using the approach developed, the non-equilibrium thermodynamics of a multicomponent gas mixture are formulated for the moment method and Burnett approximation and the boundary conditions including the slip and jump effects are derived for slightly rarefied gases. An application to transport processes in discontinuous systems and gas flows over solid bodies is also made.

A review of the vibrational relaxation of excimers is presented. Various kinetic models used in the literature to interpret experimental data and to model excimer lasers are discussed. Of these, those based on the Fokker–Planck diffusion equation are the most general due to the essentially nonequilibrium population of the hundreds of high vibrational levels and because of the multiquantum nature of the transitions involved. Numerous manifestations of these important kinetic features are analyzed. The diffusive relaxation concept has led to an overestimate of the influence of vibrational relaxation on excimer laser parameters, the saturant light flux and quantum yield expressions departing considerably from those commonly used. To gain insight into relaxation physics, considerable attention is given to the determination of kinetic parameters capable of adequately describing the electronic-vibration kinetics. Rate constants for excimer vibrational relaxation are also given.

Differences between translation-invariant and broken-symmetry bipolaron theories are analyzed in detail. It is shown that the Bogolyubov–Tyablikov canonical transformation allows collective coordinates to be introduced in a regular way for two particles in a quantum field and that for the case of the bipolaron the resulting electron–electron interaction in a phonon field depends on the electron coordinate difference alone. Predictions using a revised solution of the nonlinear differential equations for a bipolaron are given. It is shown that solving bipolaron equations numerically reduces the total bipolaron energies compared to known variational results.