Skip to main content
Open Access Publications from the University of California

UC Santa Cruz

UC Santa Cruz Previously Published Works bannerUC Santa Cruz

The evolution of radiation toward thermal equilibrium: A soluble model that illustrates the foundations of statistical mechanics


In 1916 Einstein introduced the first rules for a quantum theory of electromagnetic radiation and 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 his model to thermal equilibrium and elucidates aspects of the foundations of statistical mechanics. An analytic solution of the master equation is obtained in the Fokker-Planck approximation, which is in excellent agreement with numerical results. It is shown that the equilibrium 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. Although the counting of these configurations depends on the particle statistics, the corresponding probability is determined here by the dynamics which are embodied in 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 toward the canonical equilibrium distribution. In this limit, the present model is mathematically equivalent to an extended version of the Ehrenfests's "dog-flea" model. (C) 2004 American Association of Physics Teachers.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View