The Boltzmann equation

The problem of irreversibility came to the forefront in kinetic theory with Ludwig Boltzmann. In 1872, Boltzmann not only derived the equation that bears his name, but also introduced a definition of entropy in terms of the distribution function of the molecular velocities. By 1868, Boltzmann had already extended James Clerk Maxwell's distribution to the case where the molecules are in equilibrium in a force field with potential, including the case of polyatomic molecules. Boltzmann interprets Maxwell's distribution function in two different ways: the first way is based on the fraction of a sufficiently long time interval, during which the velocity of a specific molecule has values within a certain volume element in velocity space; whereas the second way is based on the fraction of molecules which, at a given instant, have a velocity in the said volume element. This chapter discusses the Boltzmann equation, irreversibility and kinetic theory, and Boltzmann's 1872 paper on the thermal equilibrium of gas molecules.