SOLITON STATISTICAL MECHANICS AND THE THERMALISATION OF BIOLOGICAL SOLITONS

Abstract
The calculation of the equilibrium free energy of integrable models like the sine-Gordon and attractive nonlinear Schrödinger models is discussed in the context of biological molecules like DNA : the thermalisation process (approach to equilibrium) is also discussed. The sine-Gordon model has a "repulsive" form which is the sinh-Gordon model. The approach to equilibrium of the sinh-Gordon model is described in all completeness in terms of a quantum mechanical master equation at finite temperatures. Although the dynamical evolution of the master equation as written is a solved problem, only the equilibrium solution is examined in this paper. The equilibrium free energy is calculated exactly as an integral equation for certain excitation energies at finite temperatures. Bose-fermi equivalent forms of this integral equation are given. The bose form yields a similar integral equation in classical limit. The iteration of this yields a low temperature asymptotic series for the classical free energy which checks against the result of the transfer integral method (TIM). Results for the zero temperature quantum eigenenergies are found. A further discussion of the dynamics of the approach to thermal equilbrium is made.