Paul Ehrenfest

How to use this course

Paul Ehrenfest made no single discovery on the scale of relativity or the Schrödinger equation. His gift was different. He found the weak point in an argument, built the simplest model that exposed it, and asked the question everyone else had avoided. Albert Einstein, Niels Bohr and Wolfgang Pauli treated him as the most demanding critic they had, and his students at Leiden included the discoverers of electron spin, a future Nobel laureate in economics, and several of the founders of Dutch theoretical physics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons follow his work roughly in order: statistical mechanics (1906–1912), the early quantum theory (1911–1917), relativity and dimension (1909, 1917), quantum mechanics (1927–1931) and thermodynamics (1933).

Timeline

Vienna and Boltzmann, 1880–1904

Paul Ehrenfest was born in Vienna on 18 January 1880, the youngest of five sons. His parents, Sigmund Ehrenfest and Johanna Jellinek, came from Jewish families in Moravia and ran a grocery business in the city. His mother died of cancer in 1890, and his father later married her younger sister. After his father’s death his eldest brother Arthur became his guardian. He was a sensitive, often unhappy child, aware of antisemitism at school, and his results at the Akademisches Gymnasium declined in his later years everywhere except in mathematics.

In October 1899 he entered the Technische Hochschule in Vienna to study chemistry, and he also attended lectures at the University of Vienna. There he heard Ludwig Boltzmann’s course on the mechanical theory of heat. Boltzmann’s lectures, with their mixture of rigour, physical imagination and polemic against those who doubted atoms, decided the direction of his life. The kinetic theory of gases and its foundations remained his home ground until the end.

The thesis

He took his doctorate under Boltzmann on 23 June 1904. The dissertation, Die Bewegung starrer Körper in Flüssigkeiten und die Mechanik von Hertz, examined the motion of rigid bodies in a fluid in the light of Heinrich Hertz’s mechanics, which banished forces in favour of hidden masses and constraints. Ehrenfest thought little of it and never published it in full. What he took from Vienna was a style: a physical theory had to be examined for hidden assumptions, and the best way to find them was a simple, exactly solvable model.

Göttingen, Tatyana and St Petersburg, 1901–1912

In 1901 Ehrenfest went to Göttingen, then the world centre of mathematics, and spent about eighteen months there attending lectures by Felix Klein and David Hilbert. There he met Tatyana Alexeyevna Afanasyeva, a mathematician from Kiev who had studied in St Petersburg. Women were barred from the mathematics club; Ehrenfest objected and the rule was changed. Because Austrian law did not allow a marriage between a Jew and a Christian, both renounced their religions, and they married on 21 December 1904. Their first daughter, Tatyana, born in 1905, became a mathematician in her own right.

The encyclopaedia article

Klein was editing the great Encyklopädie der mathematischen Wissenschaften. The article on the statistical foundations of mechanics had been assigned to Boltzmann. After Boltzmann’s suicide in September 1906, Klein entrusted it to the young couple. They worked on it for years, and it appeared in 1911. It became the standard critical account of what kinetic theory had and had not proved (Lesson 05).

St Petersburg

With no post available in Germany or Austria, the family moved to St Petersburg in 1907. Ehrenfest had no official position there either. He passed the Russian master’s examinations hoping for a university appointment that never came, and lived from writing and private teaching. He ran an informal seminar on modern physics, became friends with Abram Ioffe, and met the mathematicians Vladimir Steklov, Jacob Tamarkin and Alexander Friedmann. These were the years of the urn model (1907), the rotating-disk paradox (1909) and his sharp analysis of the quantum hypothesis (1911). The work was excellent and widely read, but by 1912 he was 32, unemployed and close to despair.

The call to Leiden, 1912

Early in 1912 Ehrenfest made a tour of German-speaking universities looking for work. In Prague he met Einstein for the first time, and the two became close friends at once. Einstein, about to leave Prague for Zürich, wanted Ehrenfest as his successor. The plan failed because Austrian rules required professors to declare a religious affiliation, and Ehrenfest refused to state one.

Rescue came from Leiden. Hendrik Lorentz was stepping down from the chair of theoretical physics he had held since 1878 and wanted to devote himself to research. His first choice, Einstein, had committed himself to Zürich. On the recommendation of Arnold Sommerfeld and on the strength of Ehrenfest’s published papers, Lorentz proposed him, and on 29 September 1912 a telegram announced the appointment. He arrived in October and gave his inaugural lecture, Zur Krise der Lichtäther-Hypothese (on the crisis of the light-ether hypothesis), on 4 December 1912.

Leiden in 1912 was the home of Heike Kamerlingh Onnes’s cryogenic laboratory, the coldest place on earth, and of Lorentz himself, the most respected physicist of the age. Ehrenfest felt deeply inadequate as Lorentz’s successor, a feeling he never shook off. He also started at once to build something new: a weekly colloquium, held at first in his home, where students and visitors had to explain their work in plain terms and were interrupted until they did.

The urn model and the H-theorem

For the general reader. Boltzmann claimed that entropy always increases, even though Newton’s laws run equally well backwards. Critics answered that a reversed movie of a gas is just as lawful, and that any finite system eventually returns close to its starting state. In 1907 Paul and Tatyana Ehrenfest built a toy with two urns and numbered balls that does both things at once: it relaxes toward equilibrium almost every time, and it also returns, after a time so long that no one would ever see it.

The model

Distribute \(N\) numbered balls between two urns A and B. At each step choose a number from 1 to \(N\) uniformly at random and move that ball to the other urn. Let \(n_t\) be the number of balls in A after \(t\) steps. Then \(n_t\) is a Markov chain on \(\{0,1,\dots,N\}\) with

The model was published in Physikalische Zeitschrift in 1907 under the title “On two well-known objections to Boltzmann’s H-theorem”. The objections are Josef Loschmidt’s reversibility paradox (1876) and Ernst Zermelo’s recurrence paradox (1896), the latter based on the recurrence theorem of Henri Poincaré.

Derivation

For (a), detailed balance requires \(\pi(n)\,\frac nN=\pi(n-1)\,\frac{N-n+1}{N}\), which is the identity \(n\binom Nn=(N-n+1)\binom N{n-1}\). Uniqueness follows because the chain is irreducible on a finite set. For (b), the expected change in one step is

Taking expectations gives a linear recursion with ratio \(1-2/N\). For large \(N\) the deviation decays like \(e^{-2t/N}\), which is Newton’s law of cooling in miniature: the analogue of Boltzmann’s \(H\) decreases on average.

Recurrence

For the urn model the mean time to return to the state with all balls in A is \(2^N\). With \(N=100\) and one step per second this is about \(4\times10^{22}\) years. The return to the state \(N/2\) takes only about \(\sqrt{\pi N/2}\) steps. Recurrence is certain, as Zermelo said, but for macroscopic \(N\) it is irrelevant on any human timescale. Reversibility is respected too: the chain is reversible in the sense of detailed balance, and the stationary process looks the same run forwards or backwards. The apparent arrow of time comes from starting far from equilibrium. Mark Kac solved the model completely in 1947 and called it one of the most instructive models in all of physics.

Foundations of statistical mechanics

For the general reader. Statistical mechanics replaces the impossible task of following every molecule with averages over all possible states. Why should those averages describe what one real gas actually does over time? The Ehrenfests’ 1911 encyclopaedia article asked this question with complete clarity, separated what had been proved from what had been assumed, and set the agenda for ergodic theory.

Time averages and phase averages

Let a Hamiltonian system move on the energy surface \(\Sigma_E=\{H=E\}\) with the invariant microcanonical measure \(\mu_E\) (Liouville). For an observable \(f\) define the time average along a trajectory \(x(t)\) and the phase average:

Boltzmann and James Clerk Maxwell had justified \(\bar f=\langle f\rangle_E\) by supposing that a trajectory passes through every point of the energy surface. The Ehrenfests named this the ergodic hypothesis, pointed out that a smooth one-dimensional curve cannot fill a higher-dimensional surface, and proposed instead the quasi-ergodic hypothesis: a trajectory comes arbitrarily close to every point. Shortly afterwards Arthur Rosenthal and Michel Plancherel (1913) proved the strict form impossible. The correct mathematical notion came later.

Coarse graining

Liouville’s theorem implies that the Gibbs entropy \(S=-k\int\rho\ln\rho\,d\Gamma\) of a phase-space density is exactly constant in time. The Ehrenfests insisted that the entropy which grows is a coarse-grained one: divide phase space into cells \(C_i\) of volume \(|C_i|\), let \(P_i=\int_{C_i}\rho\), and set \(S_{\mathrm{cg}}=-k\sum_iP_i\ln(P_i/|C_i|)\). A density that spreads into fine filaments keeps its fine-grained entropy, while its coarse-grained entropy rises toward the maximum. The image of ink stirred into water is theirs in spirit.

The wind-tree model

In the article they introduced a second toy. Point particles (the “wind”) move in the plane in only four directions, \(\pm\hat x,\pm\hat y\). Fixed square scatterers (the “trees”) are placed at random with their diagonals along the axes, so that a collision turns a particle through \(90^\circ\) and keeps it in the four directions. Let \(f_1,\dots,f_4\) be the numbers of particles moving in each direction. Under a Stosszahlansatz for collisions, the occupation numbers obey linear rate equations that drive them to equality, and the corresponding \(H=\sum f_i\ln f_i\) decreases. The model makes plain which step is statistical: the assumption that particles about to collide are uncorrelated with the scatterers.

The ultraviolet catastrophe and counting quanta

For the general reader. Classical physics predicted that a hot cavity should hold infinite energy, piled up at short wavelengths. In 1911 Ehrenfest gave this failure its name, the ultraviolet catastrophe, and proved something stronger: any theory that gives a finite answer and obeys Wien’s law must make energy come in discrete lumps proportional to frequency. In 1914 he and Kamerlingh Onnes gave the counting behind Planck’s formula the short proof still used today.

The catastrophe

The number of electromagnetic modes per unit volume in \([\nu,\nu+d\nu]\) is \(8\pi\nu^2d\nu/c^3\). Equipartition gives each mode mean energy \(kT\), so

This is the Rayleigh–Jeans law, after Lord Rayleigh and James Jeans. In the 1911 paper “Which features of the light quantum hypothesis play an essential role in the theory of thermal radiation?” in Annalen der Physik, Ehrenfest considered a general statistical weight for the energy of each mode. He showed that a weight compatible with Wien’s displacement law must depend on energy only through \(\varepsilon/\nu\), and that avoiding the divergence forces the weight to give finite probability to zero energy with a gap above it, a discontinuity that classical physics cannot supply. Henri Poincaré proved in 1912, independently, the stronger statement that the energies must be discrete. Together the two papers persuaded many physicists that quanta were unavoidable.

Ehrenfest later recognised that Wien’s law follows from the adiabatic invariance of \(E/\nu\) for each mode under slow compression of the cavity (Lesson 07). Planck’s rule \(E=nh\nu\) is consistent precisely because \(E/\nu\) is an adiabatic invariant.

Counting quanta (1914)

Proof. Write the \(P\) quanta as symbols \(\varepsilon\) in a row and separate them into \(N\) groups with \(N-1\) bars. Every arrangement of \(P\) symbols and \(N-1\) bars is one distribution, and conversely. Choosing which of the \(N+P-1\) positions hold symbols gives the count. ∎

With \(S=k\ln W\), Stirling’s formula and \(\partial S/\partial E=1/T\) where \(E=Ph\nu\), one obtains Planck’s mean energy per oscillator and the radiation law

The paper also stressed a conceptual point: Planck’s counting treats quanta as indistinguishable, unlike Einstein’s independent light quanta of 1905. This is the germ of Bose–Einstein statistics, made explicit by Satyendra Nath Bose in 1924.

Adiabatic invariants

For the general reader. Shorten a swinging pendulum’s string very slowly and its energy changes, but the ratio of energy to frequency stays almost exactly fixed. Ehrenfest saw that such slowly preserved quantities were the right things to quantize. If a system is in an allowed quantum state and you change it gently, it should land in another allowed state. This adiabatic principle was, with Bohr’s correspondence principle, one of the two guiding rules of the old quantum theory.

The classical theorem

The pendulum example goes back to a question Lorentz put to Einstein at the first Solvay Conference in 1911. Earlier, in 1913, Ehrenfest had noticed that an old theorem of Boltzmann and Clausius on slowly varied periodic motion was exactly what quantum theory needed.

Derivation

Along the motion \(\dot E=\partial H/\partial t=\frac{\partial H}{\partial\lambda}\dot\lambda\). Differentiating \(H(q,p(q,E,\lambda),\lambda)=E\) gives \(\partial p/\partial E=1/\dot q\) and \(\partial p/\partial\lambda=-(\partial H/\partial\lambda)/\dot q\). Hence

Averaging \(\dot E\) over one period and using \(\dot J=\frac{\partial J}{\partial E}\langle\dot E\rangle+\frac{\partial J}{\partial\lambda}\dot\lambda\) gives \(\langle\dot J\rangle=T_{\mathrm{per}}\dot\lambda\langle\partial_\lambda H\rangle-T_{\mathrm{per}}\langle\partial_\lambda H\rangle\dot\lambda=0\) to first order in \(\epsilon\). For the harmonic oscillator \(J=E/\nu\).

The adiabatic hypothesis (1916)

Starting from Planck’s oscillator, where \(E/\nu=nh\), the hypothesis transfers the quantum condition \(J=nh\) to any system that can be adiabatically connected to it. Ehrenfest used this to quantize the rotator, obtaining angular momentum \(L=n\hbar\). In 1917 his student Jan Burgers showed that for multiply periodic systems separable in suitable coordinates each action variable \(J_k=\oint p_k\,dq_k\) is an adiabatic invariant, which explained why the Bohr–Sommerfeld conditions \(J_k=n_kh\) have the form they do. Bohr adopted the principle, under the name of mechanical transformability, in his 1918 memoir on line spectra.

The quantum version is the adiabatic theorem of Max Born and Vladimir Fock (1928): a system in a nondegenerate eigenstate of \(H(\lambda)\), with a gap to the rest of the spectrum, follows the instantaneous eigenstate as \(\lambda\) changes slowly. The quantum number is preserved, exactly as Ehrenfest postulated.

Ehrenfest’s theorem and the statistics of nuclei

For the general reader. If quantum mechanics is right, why do baseballs follow Newton’s laws? In 1927 Ehrenfest showed in a one-page argument that the average position and momentum of a quantum particle obey equations very close to Newton’s. In 1931, with J. Robert Oppenheimer, he settled a puzzle about whether atomic nuclei behave as bosons or fermions, and the answer exposed a fatal flaw in the old picture of the nucleus.

Derivation

For any operator \(A\) without explicit time dependence, \(\frac d{dt}\langle A\rangle=\frac i\hbar\langle[H,A]\rangle\). With \([x,p]=i\hbar\) one finds \([p^2,x]=-2i\hbar p\) and \([V(x),p]=i\hbar V'(x)\), which give the two equations. The paper appeared in Zeitschrift für Physik in 1927 under the title “Remark on the approximate validity of classical mechanics within quantum mechanics”.

The theorem is not by itself Newton’s law, since \(\langle V'(x)\rangle\ne V'(\langle x\rangle)\) in general. Expanding about \(\langle x\rangle\),

so the centre of the wave packet follows a classical path while the packet is narrow on the scale over which the force changes. For potentials at most quadratic the correction vanishes exactly.

Statistics of composite particles (1931)

Exchanging two composites means exchanging each constituent of one with the corresponding constituent of the other. Each fermion pair contributes a factor \(-1\), so the total sign is \((-1)^{f}\) where \(f\) is the number of fermions per composite. In 1929 Franco Rasetti had found from the Raman band spectrum of \(\mathrm N_2\) that nitrogen-14 nuclei obey Bose statistics. In the proton-electron model a nitrogen-14 nucleus contains 14 protons and 7 electrons, 21 fermions, and should be a fermion. The contradiction, sharpened by Ehrenfest and Oppenheimer, was resolved when James Chadwick discovered the neutron in 1932: 7 protons and 7 neutrons make 14 fermions, a boson.

Rotating disks and three dimensions

For the general reader. Ehrenfest liked questions that look naive and turn out deep. Can a rigid wheel be spun up in special relativity? He showed in 1909 that it cannot, and the puzzle helped lead Einstein to curved geometry. Why does space have three dimensions? In 1917 he showed that in more dimensions planets could not have stable orbits and atoms could not be stable either.

The paradox (1909)

Max Born had defined a rigid body in special relativity as one whose infinitesimal neighbourhoods keep their rest lengths. Ehrenfest considered a cylinder brought from rest into uniform rotation about its axis with angular velocity \(\omega\).

The resolution is that Born rigidity is too strong. In 1910 Gustav Herglotz and Fritz Noether proved that a Born-rigid body has only three degrees of freedom, so it cannot be set rotating. Real disks deform elastically while being spun up. Seen from the co-rotating disk, rulers laid along the rim are contracted, so more of them fit around it, and the measured ratio of circumference to diameter is

Einstein used exactly this example to argue that the geometry of an accelerated frame, and therefore of a gravitational field, is non-Euclidean.

Why three dimensions (1917)

In \(n\) space dimensions, Gauss’s law for a point source gives a field strength proportional to the inverse area of a sphere, \(F(r)=-k/r^{n-1}\).

Proof. With angular momentum \(L\), the radial motion feels \(U_{\mathrm{eff}}(r)=U(r)+\frac{L^2}{2\mu r^2}\), where \(U'=k/r^m\). A circular orbit at \(r_0\) has \(k/r_0^m=L^2/(\mu r_0^3)\). Then

which is positive exactly when \(m<3\). ∎ Ehrenfest’s paper, published by the Amsterdam Academy, and a longer 1920 version in Annalen der Physik extended the comparison to atoms and to wave propagation. They are early examples of what is now called anthropic reasoning about the dimension of space.

Classifying phase transitions

For the general reader. Ice melts with a jump: you must add latent heat while the temperature stays fixed. In 1932 Leiden physicists found that liquid helium changes to a new state near 2.2 K with no latent heat at all, only a sudden change in heat capacity. Ehrenfest proposed in 1933 to classify transitions by which derivative of the free energy jumps. His scheme gave us the words first-order and second-order transition.

First order: \(S=-\partial G/\partial T\) and \(V=\partial G/\partial P\) jump, giving latent heat \(L=T\Delta S\). Equality of \(G\) on both sides along the line \(P(T)\) gives the Clausius–Clapeyron equation

The Ehrenfest relations

At a second-order transition \(\Delta S=\Delta V=0\), so Clapeyron’s formula is \(0/0\). Ehrenfest required instead that \(S\) and \(V\) remain continuous along the line. With \(\alpha=\frac1V(\partial V/\partial T)_P\), \(\kappa_T=-\frac1V(\partial V/\partial P)_T\) and heat capacity \(C_P\):

Derivation. Continuity of \(S\) along the line means \(d S_1=dS_2\), with \(dS=\frac{C_P}{T}dT-V\alpha\,dP\) by the Maxwell relation \((\partial S/\partial P)_T=-(\partial V/\partial T)_P\). Subtracting gives \(\frac{\Delta C_P}{T}dT=V\Delta\alpha\,dP\). Continuity of \(V\), with \(dV=V\alpha\,dT-V\kappa_T\,dP\), gives \(\Delta\alpha\,dT=\Delta\kappa_T\,dP\). ∎

What happened next

The motivation was the lambda point of helium, measured by Willem Keesom and his collaborators at Leiden. It turned out that the heat capacity there diverges logarithmically rather than jumping, so the transition is not second order in Ehrenfest’s strict sense. Lars Onsager’s 1944 solution of the two-dimensional Ising model showed the same kind of divergence. Modern usage distinguishes only discontinuous (first-order) and continuous transitions, described by critical exponents. Ehrenfest’s relations remain exact for transitions with finite jumps, such as the superconducting transition in zero field, where they were checked against experiment.

The Leiden school, 1912–1933

Ehrenfest made Leiden one of the great centres of theoretical physics. His weekly colloquium, which continues today as the Colloquium Ehrenfestii, was famous for its demand that every speaker be understood. Visitors to his house in Leiden signed their names on a wall upstairs; the signatures include Einstein, Bohr, Paul Dirac, Werner Heisenberg and many others. He also founded the student society De Leidsche Flesch.

His doctoral students included Jan Burgers (1918), later a founder of turbulence theory; Hendrik Kramers (1919), Bohr’s closest collaborator and Ehrenfest’s successor; Dirk Coster (1922), co-discoverer of hafnium; George Uhlenbeck and Samuel Goudsmit (both 1927); the economist Jan Tinbergen (1929), who received the first Nobel Memorial Prize in Economic Sciences in 1969; Arend Rutgers (1930); and Hendrik Casimir (1931). Enrico Fermi spent three months at Leiden in 1924 and later credited Ehrenfest’s encouragement at a time when he was unsure of himself. J. Robert Oppenheimer worked with him at Leiden in 1928, and the two later wrote a joint paper on nuclear statistics (Lesson 08).

Spin

In 1925 Uhlenbeck and Goudsmit proposed that the electron has an intrinsic angular momentum \(\hbar/2\). Ehrenfest encouraged them to write a short note and showed the idea to Lorentz, who found serious classical objections: the surface of a spinning electron would move faster than light. The students wanted to withdraw the paper, but Ehrenfest had already submitted it, and told them they were young enough to afford a foolish mistake. The paper appeared in Die Naturwissenschaften in November 1925. The objection was evaded because spin has no classical model.

Mediator between Einstein and Bohr

Einstein accepted a special professorship at Leiden in 1920, arranged by Ehrenfest and Lorentz, and stayed in the Ehrenfest house on his visits. In December 1925 Ehrenfest brought Einstein and Bohr together in Leiden, and at the fifth Solvay Conference in 1927 he was a central witness to their debate over the completeness of quantum mechanics. He was moved by Einstein’s arguments but sided with Bohr, and told Einstein so. He remained close to both.

Crisis, death and legacy

By the late 1920s Ehrenfest felt left behind by the new quantum mechanics and its formal machinery, and he judged his own abilities harshly. He suffered from recurring depression. The family also faced a heavy burden in the care of their youngest son, Wassik, born in 1918 with Down syndrome. After the Nazi seizure of power in 1933, Wassik was moved from an institution in Germany to one in Amsterdam. Ehrenfest worked hard that year to find posts for Jewish physicists dismissed from German universities.

On 25 September 1933 Ehrenfest went to the institution in Amsterdam where Wassik was being cared for, shot his son, and then shot himself. He was 53. A letter he had written to close friends some weeks earlier, and never sent, spoke of exhaustion and of feeling unable to keep up with physics. Einstein wrote a memorial essay describing him as the best teacher in their profession he had known, and as a man consumed by self-doubt. Hendrik Kramers succeeded him in the Leiden chair in 1934.

Legacy

• Statistical mechanics: the urn model is a standard first example of a reversible Markov chain and of relaxation with recurrence. The 1911 article, with Tatyana, framed the problems of ergodic theory and coarse graining for a century.
• Quantum theory: the adiabatic principle shaped the old quantum theory, and its descendants include the quantum adiabatic theorem, Berry’s phase and adiabatic quantum computation. Ehrenfest’s theorem is in every textbook.
• Relativity and thermodynamics: the Ehrenfest paradox, the dimensional argument of 1917, and the vocabulary of first- and second-order transitions.
• People: the Leiden school, which shaped Dutch theoretical physics for decades, and the style of a colloquium where clarity is compulsory.
• Honours: member of the Royal Netherlands Academy of Arts and Sciences from 1919. Prizes for work in quantum foundations and for Dutch physics theses bear his name.

Connections

People linked to Paul Ehrenfest across this collection. A name without a link has no page here yet. See Paul Ehrenfest in the network of all 337 lives →

Teachers and advisors

• Ludwig Boltzmann 1904

Students

• Arend Joan Rutgers 1930
• Dirk Coster 1922
• George Uhlenbeck 1927
• Hendrik Casimir 1931
• Hendrik Kramers 1919
• Jan Burgers 1918
• Jan Tinbergen 1929
• Samuel Goudsmit 1927

Influenced by

• David Hilbert 1901-1903
• Felix Klein 1901-1903
• Henri Poincaré

Influenced

• Enrico Fermi 1924

Colleagues and collaborators

• Arnold Sommerfeld
• Heike Kamerlingh Onnes 1914
• Hendrik Lorentz 1912-1928
• J. Robert Oppenheimer 1928-1931

Friends

• Albert Einstein 1912-1933
• Niels Bohr 1918-1933
• Abram Ioffe

Family

• Tatyana Afanasyeva 1904-1933
• Tatyana van Aardenne-Ehrenfest

Also mentioned in this life

• Max Born
• James Chadwick
• Rudolf Clausius
• Paul Dirac
• Carl Friedrich Gauss
• Josiah Willard Gibbs
• Werner Heisenberg
• Heinrich Hertz
• James Jeans
• Henri Lebesgue
• Joseph Liouville
• Andrey Markov
• James Clerk Maxwell
• Isaac Newton
• Wolfgang Pauli
• Max Planck
• Lord Rayleigh
• Erwin Schrödinger

Fields

• statistical mechanics 8 lives
• quantum theory 6 lives
• theoretical physics 11 lives

On CoursesHub

• Statistical Mechanics

Sources and further reading

Biography

1. Paul Ehrenfest, MacTutor History of Mathematics Archive.
2. Paul Ehrenfest, Wikipedia (with extensive references).
3. M. J. Klein, Paul Ehrenfest, Volume 1: The Making of a Theoretical Physicist, North-Holland, 1970.
4. M. van Lunteren and M. Hollestelle, The Delight of Thinking: The Life of Tatiana Afanassjewa and Paul Ehrenfest, Oxford University Press.
5. A. Einstein, Paul Ehrenfest: in memoriam (1934), reprinted in Out of My Later Years, 1950.
6. Lorentz Institute for Theoretical Physics, Leiden University.

Physics

1. P. and T. Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift 8 (1907) 311–314.
2. P. Ehrenfest, Gleichförmige Rotation starrer Körper und Relativitätstheorie, Physikalische Zeitschrift 10 (1909) 918.
3. P. Ehrenfest, Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?, Annalen der Physik 36 (1911) 91–118.
4. P. and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, Encyklopädie der mathematischen Wissenschaften IV.2.II (1911). English translation: The Conceptual Foundations of the Statistical Approach in Mechanics, Cornell, 1959; Dover reprint.
5. P. Ehrenfest and H. Kamerlingh Onnes, Simplified deduction of the formula from the theory of combinations which Planck uses as the basis of his radiation theory, Proc. KNAW Amsterdam 17 (1914–15).
6. P. Ehrenfest, On adiabatic changes of a system in connection with the quantum theory, Proc. KNAW Amsterdam 19 (1916) 576–597; German version in Annalen der Physik 51 (1916) 327–352.
7. P. Ehrenfest, In what way does it become manifest in the fundamental laws of physics that space has three dimensions?, Proc. KNAW Amsterdam 20 (1917).
8. P. Ehrenfest, Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik, Zeitschrift für Physik 45 (1927) 455–457.
9. P. Ehrenfest and J. R. Oppenheimer, Note on the statistics of nuclei, Physical Review 37 (1931) 333.
10. P. Ehrenfest, Phasenumwandlungen im üblichen und erweiterten Sinn, klassifiziert nach den entsprechenden Singularitäten des thermodynamischen Potentiales, Proc. KNAW Amsterdam 36 (1933).
11. M. Kac, Random walk and the theory of Brownian motion, American Mathematical Monthly 54 (1947) 369–391.
12. A. Duncan and M. Janssen, Quantization conditions, 1900–1927, arXiv:2003.04466.
13. G. Jaeger, The Ehrenfest classification of phase transitions: introduction and evolution, Archive for History of Exact Sciences 53 (1998) 51–81.