Boltzmann's Equation in Quantum Mechanics

This note gives a rigorous derivation of the quantum-mechanical analogue of Boltzmann's equation in gas theory. Assuming that the incidence of multiple encounters, other than binary encounters, between the molecules can be neglected, a closed equation for the phase space distribution of pairs of molecules is derived. The formulation of a boundary condition for this equation requires an independent appeal to statistical principles, and is the only point at which irreversibility enters the theory. A solution of the equation is given which satisfies the boundary condition. Corrections to Boltzmann's equation, which are of quantum-mechanical as well as classical origin, are derived for slightly dense gases.