Generalized phase space version of Langevin equations and associated Fokker-Planck equations

Generic Langevin equations are almost always given as first-order stochastic ordinary differential equations for the phase space variables of a system, with noise and damping terms in the equation of motion of every variable. In contrast, Langevin equations for mechanical systems with canonical position and momentum variables usually include the noise and damping forces only in the equations for the momentum variables. In this paper we derive Langevin equations and associated Fokker-Planck equations for mechanical systems that include noise and damping terms in the equations of motion for all of the canonical variables. The derivation is done by comparing a distinctive derivation of a phase space Fokker-Planck equation, given by Langer, to the usual derivation relating Langevin equations to their associated Fokker-Planck equations. The resulting equations have simple reductions to overdamped and underdamped limits. They should prove useful for numerical simulation of systems in contact with a heat bath, since they provide one additional parameter that can be used, for example, to control the rate of approach to thermal equilibrium. The paper concludes with a brief description of the modification of Kramers' result for the escape rate from a metastable well, using the new form of the Fokker-Planck equation obtained here.