# Thermal Effects: Phase-Space and Langevin Formulations

Interplay of Quantum Mechanics and NonlinearityLecture Notes in Physics（2022）

Abstract

Description of thermal effects in the dynamics of nonlinear structures occupied us in this Chapter. Questions of the thermal stability of the Davydov soliton served as a backdrop for the investigations although they were not the focus of the analysis. The conflicting situation in the literature about the fate of that soliton, originally suggested as the carrier of the energy released by the hydrolysis of adenosine triphosphate, was described as an introduction and the rest of the Chapter was dedicated not to any resolution of that issue but to provide two different modes of description, the first based on equilibrium considerations and partition function analysis and the second based on a Langevin formulation and a Fokker-Planck analysis. In the first, a magnetic analogy was utilized to set up a localization parameter which would measure the nonlinearity in the structure that develops in the system. It was shown to have a tendency to both increase and eventually decrease as the system temperature increased. Eventual Boltzmannization ensured a destruction of the nonlinear structure at high enough temperatures but the behavior at lower temperatures could be complex. The description was first developed for a two-site system interacting with a single vibration but then extended to a translationally invariant system of an arbitrary number of sites interacting with a single representative vibrational mode and then to a two-site system interacting with any number of vibrational modes. Results were shown to emerge as expected. In the second part of the Chapter, standard methods were applied to start with a Langevin picture and develop a Fokker-Planck equation. The ensuing density matrix equation was shown to be capable of describing in a unified manner, nonlinear behavior known for an adiabatic system, the onset of dissipation from non-adiabacity (finite relaxation), and even of predicting novel phenomena such as bifurcations and limit cycles. These were analyzed and it was shown how they could arise in observations of fluorescence depolarization. The first part of this Chapter, which dealt with equilibrium considerations, showed expected behavior. The second, which addressed non-equilibrium phenomena and invoked some approximations that could be considered a bit bold, presented us with effects that were surprising and certainly interesting.

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Key words

thermal effects,phase-space

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