On the derivation of the generalized Langevin equation and the fluctuation-dissipation theorem

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.