### abstract

- Here m and η are the mass of the particle and the friction coefficient respectively. Implicit is an ensemble average over realizations of the random force F. In the standard Langevin equation it represents stationary “noise” which is zero upon averaging, and whose autocorrelation function is hF (t)F(t ′ )i = φ(t−t ′ ) . (2) This phenomenological description can be derived formally from an appropriate Hamiltonian H = H0(x, p) + Henv, where the latter term incorporates the interaction with environmental degrees of freedom. The reduced dynamics of the system may be described by the propagator K(R, P |R0, P0) of the probability density matrix. For sake of comparison with the classical limit one uses Wigner function ρ(R, P) in order to represent the latter. In some cases, using Feynman-Vernon (FV) formalism [3], an exact path-integral expression for the propagator is available [4]. The FV expression is a double sum