Abstract
In 1916, Einstein rederived the blackbody radiation law of Planck that originated the idea of quantized energy one hundred years ago. For this purpose, Einstein introduced the concept of transition probability, which had a profound influence on the development of quantum theory. In this article, we adopt Einstein's assumptions with two exceptions and seek the statistical condition for the thermal equilibrium of matter without referring to the inner details of either statistical thermodynamics or quantum theory. It is shown that the conditions of thermodynamic equilibrium of electromagnetic radiation and the energy balance of thermal radiation by the matter, between any of its two energy-states, not only result in Planck's radiation law and the Bohr frequency condition, but they remarkably yield the law of the statistical thermal equilibrium of matter: the Maxwell–Boltzmann distribution. Since the transition probabilities of the modern quantum theory of radiation coincide with their definition in Einstein's theory of blackbody radiation, the presented deduction of the Maxwell–Boltzmann distribution is equally valid within the bounds of modern quantum theory. Consequently, within the framework of the fundamental assumptions, the Maxwell–Boltzmann distribution of energy-states is not only a sufficient, but a necessary condition for thermal equilibrium between the matter and radiation.
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References
M. Planck, “Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum, ” Verh. Deuts. phys. Ges. 2, 237–245 (1900).
M. Planck, “über das Gesetz der Energieverteilung im Normalspektrum, ” Ann. Physik 4, 553–566 (1901).
M. Planck, The Theory of Heat Radiation, an unabridged republication of the English tr. by M. Massius (Blakiston, 1914) from the 2nd edn. of Wärmestrahlung of 1913 (Dover, New York, 1959).
A. Einstein, “Strahlungs-emission und-absorption nach der Quantentheorie, ” Verh. Deuts. phys. Ges. 18, 318–323 (1916).
A. Einstein, “Zur Quantentheorie der Strahlung, ” Phys. Z. 18, 121–128 (1917).
A. Einstein, “On the quantum theory of radiation, ” An English tr. of his 1917 Phys. Z. article, in Sources of Quantum Mechanics, B. L. Van der Waerden, ed. (Dover, New York, 1967), pp. 63–77.
Lord Rayleigh, “Remarks upon the law of complete radiation, ” Phil. Mag. 49, 539–540 (1900).
J. H. Jeans, “On the partition of energy between matter and æther, ” Phil. Mag. 10, 91–98 (1905).
Lord Rayleigh, “The dynamical theory of gases and of radiation, ” Nature 72, 54–55 (1905).
Lord Rayleigh, “The constant of radiation as calculated from molecular data, ” Nature 72, 243–244 (1905).
P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation, ” Proc. Roy. Soc. London Ser. A 114, 243–265 (1927).
P. A. M. Dirac, The Principles of Quantum Mechanics, 4th rev. edn. (Oxford University Press, London, 1976), pp. 225–252.
W. Wien, “Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Haupsatz der Wärmetheorie, ” Sitzb. Akad. Wiss. Berlin, 55–62 (xFeb. 9, 1893).
W. Wien, “Temperatur und Entropie der Strahlung, ” Wied. Anal. 52, 132–165 (1894).
A. Sommerfeld, Thermodynamics and Statistical Mechanics, English tr. by J. Kestin, 5th pr., F. Bopp and J. Meixner, eds. (Academic, New York, 1967), pp. 145–148.
E. Schrödinger, Statistical Thermodynamics, 2nd edn. (Cambridge University Press, London, 1952), pp. 11–12 and 16.
A. Einstein and P. E. Ehrenfest, “Zur Quantentheorie des Strahlungsgleichgewichts, ” Z. Phys. 19, 301–306 (1923).
W. Pauli, “über das thermische Gleichgewicht zwischen Strahlung und freien Elecktronen, ” Z. Phys. 18, 272–287 (1923).