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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In order to count the number of microstates (called “complexions” by Boltzmann) in a model consisting of a one-dimensional molecular gas, Boltzmann introduced the idea of distributing the total energy in discrete elements of magnitude ε, but afterward he took the limit as required by classical physics. For his model consisting of Hertzian oscillators with frequency ν, Planck essentially took this discretation idea directly from Boltzmann, including his formula for the total number of configurations, but then he ignored the classical limit, and instead set the value as where is a constant.
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In November 1916 Einstein wrote to his friend Michele Besso, “A splendid idea has dawned on me about the absorption and emission of radiation…,” which led him to a new and distinct derivation of Planck’s radiation formula. See A. Pais, Subtle is the Lord: The Life and Science of Albert Einstein (Oxford U.P., New York, 1982), p. 405.
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In some textbooks, for example, C. Kittel, Elementary Statistical Physics (Wiley, New York, 1958), p. 171, it is shown that detailed balance implies equal probability of the microstates, provided that the transition probability between any pair of these states is the same in both directions. But due to the occurrence of an additional probability for spontaneous emission, this condition is not satisfied by Einstein’s theory. The reason is that Einstein’s probabilities, Eqs. (1) and (2), refer to transitions between “coarse grained” configurations which are averages over the direction of momentum of the photons.
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16.
Einstein assumed that the atoms constituted a gas in thermal equilibrium at temperature and obtained the energy distribution of radiation by the requirement that the quantum absorption and emission of radiation “does not disturb” the distribution of atomic states given by statistical mechanics.
17.
In statistical mechanics constraints such as energy conservation are introduced by the method of Lagrange multipliers. In this case, the variation of the number of atoms in the excited state and the number of photons can be treated as independent variables, while the constraint of fixed total energy of atoms and radiation, is taken into account by the well-known Lagrange multiplier
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19.
In Ref. 10 van Kampen regards as “known from equilibrium statistical mechanics,” and states that a recurrence relation like Eq. (17) for provides a connection between the transition probabilities and “which must hold if the system is closed and isolated.” Here we take the view that these transitions probabilities are known from quantum mechanics, and therefore determine as shown in Eq. (18) with , providing a dynamical justification for Boltzmann’s fundamental postulate of equal a priori probabilities.
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21.
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22.
To demonstrate the monotonic increase of total entropy in the finite temperature extension of the Ehrenfests’ model one must appeal to thermodynamics for the relation between entropy and heat exchange for systems in thermal equilibrium (Ref. 11). But this is unnecessary in our treatment of the Einstein model where the heat bath is an intrinsic part of the dynamical system.
23.
J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U.P., New York, 1955). Translated from the German by R. T. Beyer (Springer, Berlin, 1932). For a modern discussion of this quantum mechanical expression for the entropy, and the arrow of time which is determined by the direction of entropy increase,
see,
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67
, 036114
-1
036114
-6
(2003
).
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