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.
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March 2004
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March 01 2004
The evolution of radiation toward thermal equilibrium: A soluble model that illustrates the foundations of statistical mechanics
Michael Nauenberg
Michael Nauenberg
Department of Physics, University of California, Santa Cruz, California 95064
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Am. J. Phys. 72, 313–323 (2004)
Article history
Received:
May 09 2003
Accepted:
October 17 2003
Citation
Michael Nauenberg; The evolution of radiation toward thermal equilibrium: A soluble model that illustrates the foundations of statistical mechanics. Am. J. Phys. 1 March 2004; 72 (3): 313–323. https://doi.org/10.1119/1.1632488
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