Nonlinear density-matrix equation for the study of finite-temperature soliton dynamics

Abstract

The essential ideas of Davydov’s theory of molecular solitons are used to develop a variation of the Zwanzig-Nakajima projection technique. The new projection operator is time dependent and contains the feedback characteristic of the soliton dynamic. The result is a nonlinear equation of motion for the reduced density matrix of an exciton interacting cooperatively with its thermal environment. As the simplest approximation to the exact projected equation, we consider briefly a factored density operator which corresponds closely to the ansatz state vector used in wave-function treatments. This simplest approximation may be considered to be a statistical generalization of the Davydov system of evolution equations. A Markovian approximation is then considered which goes beyond the simplest approximation and contributes corrections to the Davydov-type evolution. The terms of the Markovian equation are then examined in the light of known exact results. Arguments for specific modifications of the equation are given. The resulting equation of motion embodies the correct fluctuation-dissipation relation for this system and recovers the known exact results in the transportless limit. Though the concluding equation is semiphenomenological, the equation contains no phenomenological parameters.