On the derivation of the generalized Langevin equation and the fluctuation-dissipation theorem
Hadrien Vroylandt
Abstract
Abstract The generalized Langevin equation is widely used to model the effective dynamics of chemical, soft or biological systems. It is used to describe the evolution of a small number of collective variables, and is derived using the projection operator formalism. However, the validity of the derivation of the generalized Langevin equation in systems featuring non-linear potential of mean force is presently questioned. In this paper, we rigorously derive, using a two-projection operator formalism, the usual form of the generalized Langevin equation with non-linear potential of mean force and constant memory kernel. We show that the usual fluctuation-dissipation theorem is violated and a modified version should be considered. We also illustrate this violation on a numerical example.