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Paul Ehrenfest

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Ehrenfest, Paul

Ehrenfest, Paul

(b. Vienna, Austria, 18 January 1880; d. Amsterdam, Netherlands, 25 September 1933)

theoretical physics.

Paul Ehrenfest was the youngest of the five sons of Sigmund and Johanna Jellinek Ehrenfest. His childhood was spent in a working-class district of Vienna, where his father ran a successful grocery business. He grew up surrounded by the crowded and varied life of the many nationality groups in the Austro-Hungarian capital, constantly reminded by the ugly, widespread anti-Semitism that he was a Jew. Ehrenfest’s early interest in mathematics and science was stimulated by his oldest brother, Arthur, and this fascination with science helped him through a difficult adolescence. He studied theoretical physics at Vienna, where he received his doctorate in 1904 for a dissertation on the extension of Hertz’s mechanics to problems in hydrodynamics. The dissertation was supervised by Ludwig Boltzmann, whose work and style greatly influenced Ehrenfest.

On 21 December 1904 Ehrenfest married Tatyana Alexeyevna Afanassjewa, a Russian student of mathematics whom he had met at Göttingen in 1902, during a year of study there. According to Austro-Hungarian law, the marriage of a Christian to a Jew could occur only if both partners officially renounced their religions, which Ehrenfest and his Russian Orthodox bride therefore did. During the early years of their marriage the Ehrenfests collaborated on several papers that clarified some of the obscurities in the statistical mechanics of Boltzmann and Josiah Willard Gibbs. As a result of they were invited by Felix Klein to prepare a monograph on the foundations of statistical mechanics for the Encyklopädie der mathematischen Wissenschaften. After their marriage the Ehrenfests lived first in Vienna and Göttingen and in 1907 moved to St. Petersburg, hoping to settle in Russia. Ehrenfest had no regular employment in any of these cities, but they were able to manage on their small inherited incomes. In 1911 he began a difficult and depressing search for an academic position, a search complicated by his anomalous religious status. This search came to an unexpectedly successful conclusion in 1912, when Ehrenfest was appointed to the chair of theoretical physics at Leiden as the successor to H. A. Lorentz, to whom he became deeply attached.

Ehrenfest moved to the Netherlands in October 1912 and immediately brought new vitality to the scientific life of Leiden. He started a weekly colloquium, established a reading room for physics students, revived a student science club, and generally devoted his efforts to maintaining real intellectual and human contact among all members of Leiden’s scientific community. As a teacher Ehrenfest was unique. Albert Einstein described him as “peerless” and “the best teacher in out profession whom I have ever known.” His lectures always brought out the basic concepts of a physical theory, carefully extracting them from the accompanying mathematical formalism. He worked closely with his students, doing everything in his power to help them develop their own talents. His nickname among the students, “Uncle Socrates,” captures his probing questioning, the force of a personality that could sometimes be overwhelming, and the infectious warmth of his humor.

Ehrenfest’s special gift as a theoretical physicist was his critical ability, rather than his creative power or his calculational skill. This ability is particularly evident in his writings on statistical mechanics. Boltzmann had developed this subject over a thirty-year period, and his ideas had changed a good deal during this time as he responded to a variety of difficulties pointed out to him by others. As a consequence, there was a certain amount of confusion about what his theory asserted, which assertions had been proved, and how much of the theory had survived the various attacks on it. In 1907 Ehrenfest proposed a simple theoretical model (the Ehrenfest urn model) that showed how the laws of probability could produce an average trend toward equilibrium, even though the behavior of the model was reversible in time and every one of its states would eventually recur. This meant that Boltzmann’s H-theorem (showing that molecular collisions will produce an approach to equilibrium with the entropy increasing monotonically in time), if interpreted in a suitable statistical way, did not necessarily contradict the reversible laws of mechanics, as Loschmidt had argued, or Poincaré’s recurrence theorem, as Zermelo had argued.

In their Encyklopädie article, which appeared in 1911, the Ehrenfests brought out both the logical structure and the remaining difficulties of this theory. They made a clear distinction between the older approach (before 1877), which treated the molecules statistically but tried to make universally valid statements about the gas as a whole, and the later work, in which the gas itself was treated by statistical methods. The role of the ergodic hypothesis in relating time averages to averages over an ensemble was brought out clearly; so clearly, in fact, that the attention of mathematicians was drawn to the ergodic problem. The Ehrenfests formulated the sequence of theorems that still needed proving before the statistical foundation of the second law of thermodynamics could be said to be firmly established. Their analysis of Gibbs’s approach to the subject was less sympathetic. They found fault with his treatment of irreversibility and underestimated the importance of Gibbs’s powerful ensemble methods in dealing with complex systems.

The critical approach also led Ehrenfest to his greatest positive contribution to physics: the adiabatic principle. Ehrenfest was one of the first to try to understand the significance of the strange new concept of energy quanta that Max Planck had introduced into physics in 1900 in his theory of blackbody radiation. In a series of papers culminating in his major study of 1911, “Which Features of the Quantum Hypothesis Play an Essential Role in the Theory of Heat Radiation?,” Ehrenfest picked out the essentials of the early quantum theory and showed how they fit together. He proved rigorously that the energy of electromagnetic vibrations cannot take on all values—cannot vary continuously—if the total energy of the blackbody radiation in an enclosure is to be finite: Planck’s assumption that energy is a discrete variable was, therefore, logically necessary and not just sufficient. Ehrenfest also showed, by an analysis of Wien’s displacement law, that the ratio of energy to frequency was the only variable that could be quantized for a harmonic oscillator, if one wanted to maintain the statistical interpretation of entropy. Planck’s quantum condition for the energy E of an oscillator of frequency ν,

E/ν = nh,

where n is a nonnegative integer and h is Planck’s quantum of action, no longer seemed arbitrary. The quantity E/v was the only one that kept a constant value if one varied the frequency-determining parameters sufficiently slowly, that is, adiabatically.

In a series of papers that appeared from 1913 to 1916, Ehrenfest studied the possibility of generalizing the notion of quantization, previously applied only to oscillators. He showed that every periodic system possesses a property invariant under slow (adiabatic) changes in its parameters: the ratio of the kinetic energy, averaged over one period, to the frequency. Ehrenfest proposed that only such adiabatic invariants could properly be quantized and also that when the parameters of a quantized system are changed adiabatically, the allowed quantum states of the original system continue to be the allowed quantum states. Ehrenfest and his student J. M. Burgers showed that this adiabatic principle encompassed the various quantization methods introduced independently by a variety of physicists. The adiabatic principle was widely used and highly prized as one of the few reliable guides to progress during the difficult years of the “old quantum theory,” when even the laws of conservation of energy and momentum were suspect.

During the late 1920’s and early 1930’s Ehrenfest did his utmost to try to insure the intelligibility of the new quantum mechanics and to stress its relationships with classical physics. He proved the result, still known as Ehrenfest’s theorem, that quantum mechanical expectation values of coordinates and momentum obey the classical equations of motion. One of his last papers consisted entirely of a series of fundamental questions on the physical and mathematical aspects of quantum mechanics. These questions were probably troubling many physicists, but only Ehrenfest was willing to risk the odium of asking questions that might be put aside as “meaningless.” They were far from that, as Wolfgang Pauli soon demonstrated by writing a paper answering some of Ehrenfest’s questions.

Ehrenfest affected the development of physics even more by his personal influence on other physicists than by his writings, particularly in the last decade of his life. He traveled widely, lecturing or attending conferences, and was a welcome visitor at universities from Moscow to Pasadena. When he was at home, there were always visiting physicists—older colleagues like Planck, contemporaries like Abram Fedorovitch Joffe, or young men like Enrico Fermi and Robert Oppenheimer. Ehrenfest’s way of living his physics had its effect on all who knew him, and on others through them. Both Albert Einstein and Niels Bohr were his close friends, and Ehrenfest arranged a number of the historic conversations between them on the fundamental ideas of quantum physics.

All his life Paul Ehrenfest suffered from feelings of inadequacy and inferiority. They persisted despite his extraordinary success as physicist and teacher, despite his close and warm ties to many people of many kinds. They were accentuated by the growing difficulty Ehrenfest felt in keeping up with the latest developments in his science. Finally, in September 1933, depressed by the plight of his Jewish colleagues in Nazi Germany (on whose behalf he had been exerting himself to the limit of his powers) and faced with a multitude of personal problems that seemed insuperable, Ehrenfest took his own life.


I. Original Works. Ehrenfest’s scientific papers, including his unpublished dissertation, the Encyklopädie article, and several lectures, are reprinted in his Collected Scientific Papers (Amsterdam, 1959). An English translation by M. J. Moravcsik of the Encyklpädie article appeared under the title The Conceptual Foundations of the Statistical Approach in Mechanics (Ithaca, N.Y., 1959).

Ehrenfest’s MSS, notebooks, and scientific correspondence are in the National Museum for the History of Science at Leiden. For further information, see T. S. Kuhn, J. L. Heilbron, P. Forman, and L. Allen, Sources for History of Quantum Physics (Philadelphia, 1967). pp. 33–35.

II. Secondary Literature. For a full discussion of Ehrenfest’s life and work through the period of World War I, see Martin J. Klein, Paul Ehrenfest, I, The Making of a Theoretical Physicist (Amsterdam, 1970). The second volume of this biography is in preparation. A biography in Russian by Viktor J. Frenkel has been announced but has not yet appeared.

Valuable information can be found in Albert Einstein’s essay, “Paul Ehrenfest in Memoriam,” repr. in his Out of My Later Years (New York, 1950), pp. 214–217; H. A. Kramers, “Physiker als Stilisten,” in Naturwissenschaften, 23 (1935), 297–301; Wolfgang Pauli, “Paul Ehrenfest†,” ibid., 21 (1933), 841–843; and George E. Uhlenbeck, “Reminiscences of Professor Paul Ehrenfest,” in American Journal of Physics, 24 (1956), 431–433.

Martin J. Klein

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"Ehrenfest, Paul." Complete Dictionary of Scientific Biography. . 21 Oct. 2016 <>.

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Ehrenfest, Paul

Paul Ehrenfest (poul ā´rənfĕst), 1880–1933, Austrian physicist. In 1904, Ehrenfest received his doctorate in theoretical physics in Vienna and married the Russian mathematician Tatyana Alexeyevna Afanassyewa. Together they wrote what has become a classical exposition of statistical mechanics. He was one of the first to take German physicist Max Planck's quantum theory seriously and to try to define its relation to the older physics. In 1912 he succeeded Dutch physicist H. A. Lorentz in the chair of theoretical physics at the Univ. of Leiden. He proved an energetic, lucid, and inspiring teacher. His acute criticisms and his formulation of the adiabatic principle—which Niels Bohr placed among the foundations of quantum theory—were important contributions to advancing modern physics.

See M. J. Klein, Paul Ehrenfest: The Making of a Theoretical Physicist (Vol. 1, 2d ed. 1985).

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"Ehrenfest, Paul." The Columbia Encyclopedia, 6th ed.. . 21 Oct. 2016 <>.

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"Ehrenfest, Paul." The Columbia Encyclopedia, 6th ed.. . Retrieved October 21, 2016 from