Semi-classical statistical approach to Fr\"ohlich condensation theory

TL;DR

This paper revisits Fr"ohlich phonon condensation using a semi-classical non-equilibrium statistical framework, deriving master equations and comparing with quantum models to better understand the time evolution of probability distributions.

Contribution

It introduces a semi-classical approach to Fr"ohlich condensation, deriving master equations and analyzing the probability density evolution, extending previous quantum and classical models.

Findings

Derived semi-classical master equations for phonon condensation

Established analytical links with Wu-Austin quantum Hamiltonian

Numerical solutions illustrate the condensation dynamics

Abstract

Fr\"ohlich model equations describing phonon condensation in open systems of biological relevance are here reinvestigated in a semi-classical non-equilibrium statistical context (with "semi-classical" it is meant that the evolution of the system is described by means of classical equations with the addition of energy quantization). In particular, the assumptions that are necessary to deduce Fr\"ohlich rate equations are highlighted and we show how these hypotheses led us to write an appropriate form for the master equation. As a comparison with known previous results, analytical relations with the Wu-Austin quantum Hamiltonian description are emphasized. Finally, we show how solutions of the master equation can be implemented numerically and outline some representative results of the condensation effect. Our approach thus provides more information with respect to the existing ones, in what we are concerned with the time evolution of the probability density functions instead of following average quantities.