Ludwig Boltzmann

How to use this course

Ludwig Boltzmann’s name is attached to a constant, an equation, a distribution, a law of radiation and a formula on a tombstone. Behind them lies one idea: the laws of heat are statistical laws about enormous numbers of molecules. Boltzmann pursued this idea for forty years, through mathematical difficulty, sharp objections and a long philosophical quarrel over whether atoms exist. This course presents his physics alongside his life.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. We write \(k_B\) for Boltzmann’s constant, a symbol Boltzmann himself never used: Max Planck introduced it in 1900–1901 and first wrote the entropy formula in the form \(S=k\log W\). The lessons follow Boltzmann’s work in order: equilibrium (1868–1871), the approach to equilibrium (1872), the objections and his replies (1876–1897), radiation (1884), and the debate on atoms.

Timeline

Vienna and Linz, 1844–1863

Ludwig Eduard Boltzmann was born on 20 February 1844 in Erdberg, then a suburb of Vienna. His father, Ludwig Georg Boltzmann, was an official in the imperial revenue service; his mother, Katharina Pauernfeind, came from a Salzburg merchant family. The family was Catholic and comfortably placed. The father’s postings took them to Wels and then to Linz in Upper Austria.

Boltzmann was taught at home until he was ten, then attended the Gymnasium in Linz, where he was consistently at the top of his class. He also took piano lessons, briefly from the composer Anton Bruckner, and music stayed with him all his life. His father died of tuberculosis in 1859, when Ludwig was fifteen. His mother spent much of her own money on his education. He finished school in 1863 and went to Vienna to study mathematics and physics.

Stefan’s institute, 1863–1869

At the University of Vienna, Boltzmann came under the influence of Josef Stefan, director of the Physical Institute, and of Josef Loschmidt, who in 1865 made the first reliable estimate of the size of molecules. Stefan gave him a copy of James Clerk Maxwell’s papers together with an English grammar, and the student learned the language in order to read them. Maxwell had shown in 1860 and 1867 that the velocities of gas molecules follow a definite law. Boltzmann made that law the starting point of his life’s work.

He took his doctorate under Stefan in 1866 and became his assistant. The same year he published Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie, an attempt to derive the second law of thermodynamics from mechanics alone. The attempt was premature, but the question it asked defined his career. In 1868 came the long paper Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten, which extended Maxwell’s law to molecules moving in external fields (Lesson 04). In 1869 he received the right to lecture at the university.

The young professor, 1869–1876

In 1869, on Stefan’s recommendation, Boltzmann was appointed full professor of mathematical physics at the University of Graz, at the age of 25. He used leave in the following years to work with Robert Bunsen and the mathematician Leo Königsberger in Heidelberg, and in 1871 with Gustav Kirchhoff and Hermann von Helmholtz in Berlin. The experimental skill he gained was real: in the early 1870s he measured dielectric constants of gases and solids and compared them with Maxwell’s prediction \(\varepsilon_r=n^2\), one of the early experimental supports for Maxwell’s electromagnetic theory of light.

These years produced the great theoretical papers of 1871 and 1872, including the equation that bears his name and the H-theorem (Lesson 05). In 1873 he moved to Vienna as professor of mathematics. In Graz in 1872 he had met Henriette von Aigentler, a student teacher who wanted to attend university lectures when women were not admitted; he supported her successful appeal. They married on 17 July 1876, the year he returned to Graz to take the chair of experimental physics. They had two sons and three daughters, among them Henriette (born 1880), Arthur Ludwig (1881), Ida (1884) and Else (1891). The eldest son, Ludwig, died of appendicitis in 1889, a loss that hit his father hard.

The Maxwell–Boltzmann distribution

For the general reader. In a gas at rest, molecules move in all directions at many speeds. Maxwell found the law that says what fraction have each speed. Boltzmann generalized it: in any system in equilibrium, a state of energy \(E\) is occupied with a weight proportional to \(e^{-E/k_BT}\). This single exponential explains why air thins with altitude, why chemical reactions speed up with heat, and much else.

Maxwell’s argument and Boltzmann’s

Maxwell (1860) assumed that the three velocity components are independent and that the distribution depends only on speed: \(\phi(v_x)\phi(v_y)\phi(v_z)=F(v_x^2+v_y^2+v_z^2)\). The only solutions are Gaussians, \(\phi\propto e^{-\beta mv_x^2/2}\). Independence was an assumption, which Maxwell himself later found unsatisfactory. Boltzmann’s 1868 paper derived the distribution instead from the dynamics of collisions and showed it holds also in external force fields and for molecules with internal structure. His 1872 paper (Lesson 05) showed it is the unique stationary state of the collision process.

Speed distribution and moments

Integrating over directions gives the speed distribution \(P(v)=4\pi v^2\left(\frac{m}{2\pi k_BT}\right)^{3/2}e^{-mv^2/2k_BT}\). Its most probable, mean and root-mean-square speeds are

and the mean kinetic energy is \(\tfrac32k_BT\), the kinetic definition of temperature.

The Boltzmann equation and the H-theorem

For the general reader. Showing that Maxwell’s distribution is an equilibrium is not enough: why does a gas started in any state end up there? In 1872 Boltzmann wrote an equation for how the velocity distribution changes as molecules collide, and found a quantity, which he later called \(H\), that can only decrease. It stops decreasing exactly when the gas reaches Maxwell’s distribution. Here, for the first time, the irreversibility of nature was derived from a model of molecules.

The equation

Let \(f(\mathbf x,\mathbf v,t)\) be the number density of molecules in phase space. For a dilute gas of particles with binary elastic collisions and external force \(\mathbf F\),

Here \(f_1=f(\mathbf v_1)\), primes denote velocities after a collision with scattering angle \(\Omega\), and \(\sigma\) is the differential cross-section. The right side counts collisions that put molecules into velocity \(\mathbf v\) minus those that take them out. Writing the rate of collisions as proportional to the product \(ff_1\) assumes that the velocities of two molecules about to collide are uncorrelated. Boltzmann later called this assumption molecular disorder; the name Stosszahlansatz (collision-number hypothesis) became standard through the 1912 article of Paul and Tatiana Ehrenfest.

Proof

Since \(\int\partial_tf\,d^3v=0\), we have \(\dot H=\int(1+\ln f)\,\partial_tf\,d^3v=\int\ln f\,C[f]\,d^3v\). Write \(dw\) for the collision measure \(|\mathbf v-\mathbf v_1|\sigma\,d\Omega\,d^3v\,d^3v_1\). Interchange \(\mathbf v\leftrightarrow\mathbf v_1\) (symmetry of the collision), and then unprimed \(\leftrightarrow\) primed (invariance of the measure for elastic collisions, a consequence of microscopic reversibility). Averaging the four expressions,

because \((a-b)\ln(b/a)\le0\) for positive \(a,b\). Equality requires \(ff_1=f'f'_1\) for all collisions, so \(\ln f\) is additive in collisions, hence a combination of the conserved quantities: \(\ln f=a+\mathbf b\cdot\mathbf v-cv^2\). For an ideal gas, \(S=-k_BH\cdot V+\text{const}\), so the H-theorem is the second law for this model.

Reversibility and recurrence

For the general reader. Newton’s laws work equally well run backwards. How, then, can a theorem built on them prove that a quantity always decreases? Boltzmann’s teacher and friend Josef Loschmidt pointed this out in 1876. Twenty years later Ernst Zermelo added a second objection: a closed mechanical system returns arbitrarily close to its starting state. Boltzmann’s answers turned the H-theorem from a theorem of mechanics into a statement about probability.

Loschmidt (1876)

If a trajectory takes the gas from state \(A\) to state \(B\) with \(H\) decreasing, reversing all velocities in \(B\) gives a trajectory of the same mechanics along which \(H\) increases. So no theorem of mechanics alone can make \(H\) decrease for every initial state. In 1877 Boltzmann replied that the states from which \(H\) increases, though they exist, are overwhelmingly outnumbered: almost every state leads toward equilibrium. The hidden assumption is molecular chaos, which holds for colliding pairs before the collision but not after it.

Proof

Let \(B\subset A\) be the points that never return. The sets \(B,T^{-1}B,T^{-2}B,\dots\) are pairwise disjoint: if \(x\in T^{-i}B\cap T^{-j}B\) with \(i<j\), then \(y=T^ix\in B\) and \(T^{j-i}y\in B\subset A\), contradicting the definition of \(B\). They all have measure \(\mu(B)\), and their union has measure at most 1, so \(\mu(B)=0\). Applying this to \(T^n\) gives infinitely many returns.

Zermelo used this theorem in 1896 to argue that \(H\) cannot decrease monotonically. Boltzmann answered that the recurrence times for a macroscopic gas are absurdly long, far longer than the age of the universe, so the theorem is irrelevant to observation. He also offered a cosmological speculation: perhaps the universe as a whole is in equilibrium, and our region is a rare, huge fluctuation, in which time’s arrow points toward increasing entropy wherever living beings exist.

Entropy and probability, 1877

For the general reader. Shuffle a deck of cards and it is almost never in order, because there is only one ordered arrangement and an astronomical number of disordered ones. Boltzmann saw that entropy measures exactly this: the number of microscopic arrangements compatible with what we observe. Equilibrium is simply the most numerous kind of state. The formula \(S=k\log W\), cut into his gravestone, sums up the idea.

The counting argument

In Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung (1877), Boltzmann divided the energy of each molecule into discrete cells \(0,\varepsilon,2\varepsilon,\dots\) as a mathematical device, to be refined to zero at the end. A distribution, which he called a Komplexion class, specifies how many molecules \(n_i\) have energy \(\varepsilon_i\). The number of ways to realize it with \(N\) distinguishable molecules is

Derivation

With Stirling’s formula \(\ln n!\approx n\ln n-n\), \(\ln W\approx N\ln N-\sum_in_i\ln n_i\). Maximize with Lagrange multipliers \(\alpha,\beta\): \(\partial_{n_i}\big[\ln W-\alpha\sum n_i-\beta\sum n_i\varepsilon_i\big]=-\ln n_i-1-\alpha-\beta\varepsilon_i=0\), so \(n_i=e^{-1-\alpha}e^{-\beta\varepsilon_i}\). Writing \(p_i=n_i/N\), \(\ln W\approx-N\sum_ip_i\ln p_i\), which is \(-H\) of Lesson 05 in discrete form, and later the Shannon entropy. Comparison with \(dS=dQ/T\) identifies \(\beta=1/k_BT\).

Boltzmann did not write \(S=k\log W\) in this form, nor did he use the constant \(k\). Planck did both in his work on black-body radiation in 1900–1901, using Boltzmann’s counting with finite energy elements \(\varepsilon=h\nu\), which he did not let go to zero. Boltzmann’s mathematical device became the quantum.

Equipartition, ergodicity, ensembles

For the general reader. In equilibrium, energy is shared equally among all the independent ways a molecule can move. This rule, the equipartition theorem, explains the heat capacities of gases and solids. Its failures, for some gases and later for light, were among the first signs that classical physics was incomplete. Boltzmann also asked why time averages should equal averages over all states, and in doing so founded ergodic theory.

Proof. Integrate by parts in \(z_j\): \(\int z_i\,\partial_jH\,e^{-H/k_BT}dz=-k_BT\int z_i\,\partial_j\big(e^{-H/k_BT}\big)dz=k_BT\,\delta_{ij}\int e^{-H/k_BT}dz\). For \(H=\tfrac12az_i^2+\cdots\), \(z_i\partial_iH=2\cdot\tfrac12az_i^2\), giving \(\langle\tfrac12az_i^2\rangle=\tfrac12k_BT\).

The diatomic puzzle

Measured ratios of specific heats \(\gamma=c_p/c_v\) for air and hydrogen are near \(1.4\). In 1876 Boltzmann showed this follows if a diatomic molecule moves like a rigid dumbbell: three translational and two rotational quadratic terms give \(c_v=\tfrac52k_B\) per molecule and \(\gamma=7/5\). But a real molecule should also vibrate, and real atoms have internal parts, which classical equipartition says must share the energy. Why they did not was a puzzle Boltzmann could not resolve. Quantum theory resolves it: modes with level spacing much larger than \(k_BT\) are frozen.

Ergodicity and ensembles

In 1868 and 1871 Boltzmann assumed that a single isolated system, over a long time, passes through every state on its energy surface, so that time averages equal phase-space averages. The Ehrenfests later named this the ergodic hypothesis. In its literal form it is impossible for a smooth flow on a manifold of dimension greater than one, and the modern ergodic theorems of John von Neumann and George Birkhoff (1931–1932) replace it with metric indecomposability. In 1884 Boltzmann also introduced families of systems in different states, which he called monodes and ergodes; these are the canonical and microcanonical ensembles that Josiah Willard Gibbs developed systematically in 1902.

The Stefan–Boltzmann law

For the general reader. A hot body glows, and the total power it radiates rises steeply with temperature. In 1879 Josef Stefan inferred from experiments that the power grows as the fourth power of the absolute temperature. In 1884 his former student showed why: light in a cavity behaves like a gas with pressure, and the laws of thermodynamics then force the fourth power. It was one of the first applications of thermodynamics to radiation, and it prepared the way for Planck.

Derivation

For radiation in volume \(V\), \(U=u(T)V\). The thermodynamic identity \(\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial p}{\partial T}\right)_V-p\) gives

The pressure \(p=u/3\) follows from Maxwell’s electromagnetic theory: each photon direction contributes momentum flux, and averaging \(\cos^2\theta\) over directions gives one third. The constant \(a\) cannot be found by thermodynamics. Planck’s law gives \(a=\frac{8\pi^5k_B^4}{15h^3c^3}\), so \(\sigma=\frac{2\pi^5k_B^4}{15h^3c^2}=5.67\times10^{-8}\ \mathrm{W\,m^{-2}K^{-4}}\).

Fluctuations and the reality of atoms

For the general reader. If entropy is only overwhelmingly likely to increase, then small systems should show small, random departures from equilibrium. Boltzmann’s statistical view predicted such fluctuations; the energetics school, which treated thermodynamics as exact, did not. Within a few years of his death, the jittering of tiny particles in water, explained by Albert Einstein and measured by Jean Perrin, gave the decisive evidence that atoms are real.

Proof. With \(Z=\int e^{-\beta H}\), \(\langle E\rangle=-\partial_\beta\ln Z\) and \(\langle\Delta E^2\rangle=\partial_\beta^2\ln Z=-\partial_\beta\langle E\rangle=k_BT^2\,\partial_T\langle E\rangle\). Since \(C_V\) and \(\langle E\rangle\) are both proportional to \(N\), relative fluctuations scale as \(N^{-1/2}\): invisible for a mole, important for a colloidal particle.

Brownian motion

A sphere of radius \(r\) in a fluid of viscosity \(\eta\) feels drag \(6\pi\eta r\,v\). Einstein showed in 1905 that its mean square displacement along one axis grows as \(\langle x^2\rangle=2Dt\) with

Measuring \(D\) therefore measures \(k_B\), and with the gas constant \(R\), Avogadro’s number \(N_A=R/k_B\). Jean Perrin’s experiments of 1908–1909 gave values consistent with those from the kinetic theory of gases and from radiation. Ostwald accepted the existence of atoms in 1909. Boltzmann had not lived to see it.

Graz, Munich, Vienna, Leipzig, 1876–1902

Boltzmann’s second period in Graz, from 1876 to 1890, was the most settled of his life. He directed the physics institute and did experiments as well as theory. Svante Arrhenius and Walther Nernst both worked with him in Graz as young researchers. In 1885 he was elected to the Imperial Austrian Academy of Sciences, and in 1887 he served as rector of the university. In 1888 he was offered Kirchhoff’s chair in Berlin, and he declined it.

From 1890 he held the new chair of theoretical physics at Munich, a productive and happy period. On Stefan’s death he returned in 1894 to Vienna as professor of theoretical physics. In 1895 Ernst Mach was appointed to a chair of philosophy at Vienna. Mach held that science should describe sensations and their relations, and regarded atoms as at best a useful fiction. At a meeting in Lübeck the same year, Georg Helm and Wilhelm Ostwald argued that energy, not matter, was the basic reality and that thermodynamics needed no molecules. Boltzmann attacked their arguments forcefully, and most of the physicists present sided with him. The debate continued for a decade in print, and in his preface to the second volume of the Gastheorie (1898) Boltzmann described himself as an individual struggling weakly against the stream of the times.

In 1900 he moved to Leipzig, where Ostwald, his scientific opponent but a personal friend, had invited him. He was unhappy there and suffered severe depression. In 1902 he returned to his old chair in Vienna, which had been left vacant for him.

Last years, death and legacy, 1902–1906

Back in Vienna, Boltzmann took over in 1903 the course on the philosophy of nature that Mach had given until ill health forced his retirement. The lectures were a public success; crowds filled the hall. The same year he helped found the Austrian Mathematical Society. In 1904 he spoke at the Congress of Arts and Science at the World’s Fair in St Louis, and in 1905 he taught a summer course at the University of California in Berkeley, which he described in a humorous travel essay, Reise eines deutschen Professors ins Eldorado. His doctoral students in these years included Paul Ehrenfest (1904) and Philipp Frank; Lise Meitner attended his lectures and later spoke of their lasting influence on her.

His health, eyesight and spirits were failing. In May 1906 he gave up teaching; the dean’s note spoke of a serious form of neurasthenia. During a summer holiday with his wife and daughter at Duino, on the Adriatic coast near Trieste, he took his own life on 5 September 1906. He was 62. The popular story that he was driven to despair by the rejection of his ideas is not supported by historians; his illness had a long history. He is buried in the Zentralfriedhof in Vienna, under a bust and the formula \(S=k\log W\).

Legacy

• Statistical mechanics: the Boltzmann factor, the probabilistic interpretation of entropy, equipartition, and ensembles are the foundation of the subject, completed by Gibbs and Einstein.
• Kinetic theory: the Boltzmann equation governs dilute gases, plasmas, neutron transport, semiconductors and radiative transfer; its derivation from mechanics for short times was proved by Oscar Lanford in 1975.
• Quantum theory: Planck’s 1900 derivation of the radiation law used Boltzmann’s counting with finite energy elements, the first step toward the quantum.
• Mathematics: ergodic theory, large deviations, and information theory all grew from his questions; \(-\sum p\ln p\) is his \(H\).
• Honours: Foreign Member of the Royal Society (1899), member of the Austrian and Swedish academies. The IUPAP Boltzmann Medal for statistical physics has been awarded since 1975.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Ludwig Boltzmann, MacTutor History of Mathematics.
2. Ludwig Boltzmann, Wikipedia (with references).
3. J. Uffink, Boltzmann’s Work in Statistical Physics, Stanford Encyclopedia of Philosophy.
4. C. Cercignani, Ludwig Boltzmann: The Man Who Trusted Atoms, Oxford University Press, 1998.
5. E. Broda, Ludwig Boltzmann: Man, Physicist, Philosopher, Ox Bow Press, 1983.

Physics

1. L. Boltzmann, Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten, Wiener Berichte 58 (1868).
2. L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Wiener Berichte 66 (1872). English translation in S. G. Brush, Kinetic Theory, vol. 2, Pergamon, 1966.
3. L. Boltzmann, Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, Wiener Berichte 76 (1877). English translation by K. Sharp and F. Matschinsky, Entropy 17 (2015).
4. L. Boltzmann, Ableitung des Stefan’schen Gesetzes, Annalen der Physik 22 (1884).
5. L. Boltzmann, Vorlesungen über Gastheorie, 2 vols., Barth, Leipzig, 1896–1898; English translation Lectures on Gas Theory, Dover.
6. P. and T. Ehrenfest, The Conceptual Foundations of the Statistical Approach in Mechanics (1912), Dover reprint.
7. H. R. Brown, W. Myrvold and J. Uffink, Boltzmann’s H-theorem, its limitations, and the birth of (fully) statistical mechanics, arXiv:0809.1304.
8. C. Cercignani, The Boltzmann Equation and Its Applications, Springer, 1988.