Hard sphere Langevin equations

The Zwanzig projection operator formalism is used to obtain Langevin equations for fluctuations of the microscopic single particle phase space density about its equilibrium value. For hard sphere dynamics the autocorrelation function of the fluctuating force has a singular contribution. The origin of this singular part is identified from an identity relating hard sphere binary collision operators for positive and negative time evolution. Langevin equations with white noise are obtained in two limits. The first is low density, leading to the Boltzmann-Langevin equation; the second is short times, leading to the Enskog-Langevin equation.