Ludwig Boltzmann

Major contributions

• Generalised Maxwell's velocity distribution into the Maxwell–Boltzmann distribution for gases in external fields, and introduced the Boltzmann factor e^−E/kT that gives the relative probability of states of energy E
• Formulated the Boltzmann transport equation (1872) for the evolution of a dilute gas, and the H-theorem, which shows that a quantity related to entropy decreases (entropy increases) as molecules collide
• In 1877 related entropy to the number of microscopic states compatible with a macroscopic state, the relation now written S = k_B ln W (see Entropy)
• Derived in 1884, from thermodynamics and Maxwell's radiation pressure, the T⁴ law for black-body radiation that his teacher Josef Stefan had found empirically (the Stefan–Boltzmann law)
• Defended atomism and the kinetic theory against the “energeticists” (notably Wilhelm Ostwald) and the positivist objections of Ernst Mach

Named concepts

Concept	Statement	Codex page
Boltzmann constant (kB)	Links energy and temperature; exactly 1.380649×10−23 J/K since 2019	View entry
Boltzmann entropy	S = kB ln W	View entry
Stefan–Boltzmann law	Radiated power per area ∝ T4	View entry
Boltzmann equation / H-theorem	Kinetic evolution of a gas towards equilibrium	View entry

Career

Boltzmann was born in Vienna and studied at the University of Vienna, receiving his doctorate in 1866 under Josef Stefan. He held professorships at Graz (1869–1873 and 1876–1890), Vienna (1873–1876, 1894–1900 and from 1902), Munich (1890–1894) and Leipzig (1900–1902). He was a respected lecturer whose students included Lise Meitner and Paul Ehrenfest, and he lectured on philosophy as well as physics.

His statistical arguments drew sharp objections. Josef Loschmidt (1876) argued that time-reversible mechanics cannot produce irreversible entropy increase (the reversibility paradox), and Ernst Zermelo (1896) cited Poincaré's recurrence theorem. Boltzmann's answer, that entropy increase is overwhelmingly probable rather than certain and depends on a low-entropy starting state, is the basis of the modern understanding (see Second Law). Boltzmann suffered from what is now generally described as bipolar disorder or severe depression, and he died by suicide in 1906 while on holiday near Trieste. His grave in Vienna's Zentralfriedhof bears the formula S = k log W.

Key publications

• Boltzmann, L. (1872). “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen.” Wiener Berichte, 66, 275–370.
• Boltzmann, L. (1877). Wiener Berichte, 76, 373–435.
• Boltzmann, L. (1896–1898). Vorlesungen über Gastheorie (Lectures on Gas Theory). Leipzig: J.A. Barth.

Nobel Prize

Boltzmann never received a Nobel Prize; the prizes began in 1901, five years before his death, and statistical mechanics was not recognised by the committee in that period. The constant k_B that bears his name was introduced in that explicit form by Max Planck in 1900, who received the 1918 prize partly on the basis of Boltzmann's statistical methods (see Max Planck).

Historical context

Within a few years of Boltzmann's death, the reality of atoms was settled: Einstein's 1905 theory of Brownian motion, using statistical-mechanical reasoning, was confirmed by Jean Perrin's experiments (1908–1909), and Ostwald accepted atomism. Boltzmann's methods became the foundation of quantum statistics (see Fermi Energy and Bose-Einstein Condensation). It is often suggested that the hostile reception of his ideas contributed to his despair; historians caution that his illness had many causes and that this link cannot be established. This page records the circumstances of his death factually and without speculation.