Anomalous motion generated by the Coulomb friction in the Langevin equation

We study the stability of the Maxwell–Boltzmann (i.e., Gaussian) distribution for the density of states at equilibrium, against an arbitrary choice of the friction in the Langevin equation. We find that this distribution is Gaussian, if and only if the friction is Lipschitz continuous. In particular, we argue that the origin of the exponential (instead of Gaussian) velocity distribution (PDF) of particles when the viscous friction is replaced by the Coulomb friction in the Langevin equation with white noise is due to the non-Lipschitz continuity of the Coulomb friction, a feature of solid friction. The use of the Fokker–Planck equation to determine the exponential PDF is justified, since the subset on which the friction is not continuous is of zero probability (v=0). The application to the motion of granular gases is discussed.