Classical and quantum dissipation in non-homogeneous environments

We investigate a generalization of the oscillator model of a particle interacting with a thermal reservoir, in which arbitrary nonlinear couplings in the particle coordinates are present. The equilibrium positions of the heat bath oscillators are promoted to space-time functions, which are shown to represent a modulation of the internal noise by the external forces. The model thus provides a description of classical and quantum dissipation in non-homogeneous environments. In the classical case we derive a generalized Langevin equation with nonlinear multiplicative noise and a position-dependent fluctuation-dissipation theorem associated to non-homogeneous dissipative forces. When time-modulation of the noise is present, a new force term is predicted besides the dissipative and random ones. The model is quantized to obtain the non-homogeneous influence functional and master equation for the reduced density matrix of the Brownian particle. The quantum evolution equations reproduce the correct Langevin dynamics in the semiclassical limit. The consequences for the issues of decoherence and localization are discussed.