Ludwig Boltzmann

Ludwig Eduard Boltzmann (February 20, 1844 to September 5, 1906) was an Austrian physicist who founded statistical mechanics. He showed how the laws of heat follow from the motion of atoms and molecules too small to see, and he argued that the second law of thermodynamics is a statement of probability: a disordered state is overwhelmingly more likely than an ordered one because there are far more ways to be disordered.^[1]

The formula that relates entropy to the logarithm of the number of microscopic arrangements is cut into his tombstone in Vienna, though Uffink notes that he "never actually wrote this formula down".^[1] The constant that carries his name converts temperature into energy. Every calculation of thermal noise contains it, and since 2019 its value has defined the kelvin.^[2]

Boltzmann was born in Vienna, where his father was a taxation official, and took his doctorate at the University of Vienna in 1866 under Josef Stefan, with a thesis on the kinetic theory of gases.^[3] In 1869 he became professor of theoretical physics at Graz. Four years later he went to Vienna as professor of mathematics, and three years after that he was back in Graz in the chair of experimental physics. He said, at least half jokingly, that he moved so much because he had been born in the last hours of a Mardi Gras ball.^[3]

Munich, Vienna and Leipzig bid for him in turn, in 1889, 1894 and 1900, sometimes pooling the salaries of several professorships to do it. In 1888 he had declined, after a curious sequence of negotiations, the chair in Berlin that Gustav Kirchhoff had left.^[1] He went back to Vienna in 1902, to the chair of theoretical physics and to the philosophy course that Ernst Mach had given until his retirement in 1901. The lectures became so popular that the largest hall available could not hold the audience.^[3] In the summer of 1905 he gave lectures at the University of California, Berkeley, and wrote up the trip as "Reise eines deutschen Professors ins Eldorado", which is printed in his Populäre Schriften of that year.^[4] Physics Today, which printed a translation as "A German professor's trip to El Dorado" in 1992, summarizes it as an account of a summer spent smuggling wine into Berkeley and dining at the elbow of William Randolph Hearst's mother.^[5]

Boltzmann's first paper on the statistical theory of heat, in 1866, aimed to derive the second law from mechanics alone. Within two years he had read James Clerk Maxwell's papers on gas theory of 1860 and 1867, in which the state of a gas is described by a probability distribution of molecular velocities.^[1] Boltzmann's papers of 1868 and 1871 extended Maxwell's distribution to gases in a static external force and to molecules made of several atoms. The result, the distribution of energies among molecules at a given temperature, is now called the Maxwell-Boltzmann distribution, and the factor exp(−E/kT) in it is the Boltzmann factor.^[1]

In 1872 he published the paper that contains the Boltzmann equation, which describes how the distribution changes as molecules collide, and the H-theorem, which says that a quantity H computed from the distribution can only decrease until the distribution is Maxwell's. He took this to be the mechanical counterpart of the second law.^[1] Josef Loschmidt, his former teacher and lifelong friend, objected in 1876 that the equations of motion are reversible in time, so a theorem that runs in one direction cannot follow from them alone. Ernst Zermelo added in 1896 that a closed mechanical system must eventually return close to its starting state.^[1] Boltzmann's reply, in 1877, was to turn to probability. A gas in an improbable state will almost always move to more probable states, and the equilibrium state is the most probable of all, because it can be realized in overwhelmingly more ways.^[1]

That paper of 1877 counts. It takes a macroscopic state of a gas, such as a given number of molecules in each half of a box, and counts the microscopic arrangements that produce it.^[1] The state with the molecules evenly split has by far the most. For 100 molecules in two halves, one arrangement puts all of them on the left and 1.0 × 10²⁹ of them put exactly 50 on each side.^[a] With a number of molecules near a mole the lopsidedness is total, and the second law becomes a certainty in practice.

The logarithm turns the product of independent counts into a sum, which is what an entropy must do. Max Planck wrote the relation with a constant of proportionality k in his derivation of the law of black-body radiation, in the paper of 1901 that followed his first announcement of the law late in 1900: the entropy S equals k log ℜ, where ℜ is the number of "Complexionen", a word that Planck says he took from Boltzmann.^[6] Planck's value for the constant was 1.346 × 10⁻²³ J/K, about 2.5 percent below the modern figure. Before 1900, equations with Boltzmann factors were written with the gas constant R and macroscopic energies. The gas constant is k multiplied by the Avogadro constant, and since 2019 both have been exact.^[2]

In 1884 Boltzmann showed that Stefan's law of 1879, that the energy radiated by a hot body rises as the fourth power of its absolute temperature, could be derived from thermodynamics and Maxwell's electromagnetic theory.^[3][7] The law is the Stefan-Boltzmann law.

Boltzmann defended the atomic picture when Ernst Mach and Wilhelm Ostwald doubted it. In his account of the dispute he said of the atoms that they gave at best an analogy, a picture of reality, and that any approach to natural science that avoided hypotheses entirely would fail.^[1] Jean Perrin's book Atoms credits "men such as Dalton and Boltzmann" with having given science the doctrine of atoms, the method of trying "to explain the complications of the visible in terms of invisible simplicity".^[8] The argument over atoms, and the claim that the dispute drove Boltzmann to suicide, are treated in Molecular reality (history of science).

Boltzmann died by suicide at Duino, near Trieste, on September 5, 1906.^[3][1] Uffink finds "no factual evidence" that he was ignored or lacked recognition, and attributes his death to factors in his personal life, "a decline of mental and physical health".^[1] He is buried in the Zentralfriedhof in Vienna.

The distribution of ions near a charged pore wall follows the Boltzmann factor, and the width of the cloud is the Debye length, set by kT against the electrostatic energy (see Electric double layer). The thermal voltage kT/e is 25.7 mV at 25 °C, so the 120 mV commonly applied across a pore is about 4.7 times the thermal energy per charge. Johnson-Nyquist noise gives the pore its noise floor. The mean square noise current of a resistance R in a bandwidth B is 4kTB/R, so a pore of 1 GΩ in 10 kHz has 0.41 pA of rms noise, against an open-pore current near 120 pA.^[2]

Maxwell's demon, a thought experiment of 1867 that sorts fast molecules from slow ones, is a puzzle about the statistical reading of the second law; see Maxwell's demon.

• James Clerk Maxwell
• Ernst Mach
• Brownian motion
• Maxwell's demon
• Jean Perrin