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

Generic Langevin equations are almost always given as first-order stochasti c 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 posi tion and momentum variables usually include the noise and damping forces on ly in the equations for the momentum variables. In this paper we derive Lan gevin equations and associated Fokker-Planck equations for mechanical syste ms that include noise and damping terms in the equations of motion for all of the canonical variables. The derivation is done by comparing a distincti ve 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 overda mped and underdamped limits. They should prove useful for numerical simulat ion of systems in contact with a heat bath, since they provide one addition al parameter that can be used, for example, to control the rate of approach to thermal equilibrium. The paper concludes with a brief description of th e modification of Kramers' result for the escape rate from a metastable wel l, using the new form of the Fokker-Planck equation obtained here.