The evolution of radiation towards thermal equilibrium: A soluble model which illustrates the foundations of Statistical Mechanics

TL;DR

This paper extends Einstein's quantum radiation model to include stochastic dynamics, deriving a master equation that describes how systems approach thermal equilibrium, and provides an analytic solution aligning with numerical results.

Contribution

It introduces a time-dependent stochastic framework for Einstein's radiation model, deriving a master equation and solving it analytically within the Fokker-Planck approximation.

Findings

Analytic solution matches numerical simulations

Probability distribution aligns with Boltzmann's postulate

Photon bath can drive atoms to Gibbs equilibrium

Abstract

In 1916 Einstein introduced the first rules for a quantum theory of electromagnetic radiation, and he applied them to a model of matter in thermal equilibrium with radiation to derive Planck's black-body formula. Einstein's treatment is extended here to time-dependent stochastic variables, which leads to a master equation for the probability distribution that describes the irreversible approach of Einstein's model towards thermal equilibrium, and elucidates aspects of the foundation of statistical mechanics. An analytic solution of this equation is obtained in the Fokker-Planck approximation which is in excellent agreement with numerical results. At equilibrium, it is shown that the probability distribution is proportional to the total number of microstates for a given configuration, in accordance with Boltzmann's fundamental postulate of equal a priori probabilities for these states. While the counting of these configurations depends on particle statistics- Boltzmann, Bose-Einstein, or Fermi-Dirac - the corresponding probability is determined here by the dynamics which are embodied in the form of Einstein's quantum transition probabilities for the emission and absorption of radiation. In a special limit, it is shown that the photons in Einstein's model can act as a thermal bath for the evolution of the atoms towards the canonical equilibrium distribution of Gibbs. In this limit, the present model is mathematically equivalent to an extended version of the Ehrenfests' ``dog-flea'' model, which has been discussed recently by Ambegaokar and Clerk.