External potential modifies memory of solute particles: A particle-viscous bath model

Abstract

The response to external forces is one of the important properties of stochastic processes. In the present paper, we derive the generalized Langevin equation (GLE) for the description of the non-Markovian Brownian motion (BM) of a particle, e.g. a molecular solute in a solvent, in a harmonic potential. It is assumed that not only the solute particle but also the surrounding molecules respond to the external field. This effect is taken into account by a modification of the Caldeira-Legget particle-bath model. It is also assumed that the bath particles experience friction, modeled by the Stokes force. We show that under the influence of the external force the memory function and the random thermal force are affected by the external potential. Explicit expressions for these functions and the elastic force acting on the Brownian particle are obtained in the discrete variant of the model and its continuum approximation. In the latter case, analytical formulas for relevant time correlation functions describing the dynamics of the solute are obtained coming from the memory function that describes its free BM.