On the derivation of a Nonlinear Generalized Langevin Equation

We recast the Zwanzig's derivation of a nonlinear generalized Langevin equation (GLE) for a heavy particle interacting with a heat bath in a more general framework. We show that it is necessary to readjust the Zwanzig's definitions of the kernel matrix and noise vector in the GLE in order to recover the correct definition of fluctuation-dissipation theorem and to be able performing consistently the continuum limit. As shown by Zwanzig, the nonlinear feature of the resulting GLE is due to the nonlinear dependence of the equilibrium map by the heavy particle variables. Such an equilibrium map represents the global equilibrium configuration of the heat bath particles for a fixed (instantaneous) configuration of the system. Following the same derivation of the GLE, we show that a deeper investigation of the equilibrium map, considered in the Zwanzig's Hamiltonian, is necessary. Moreover, we discuss how to get an equilibrium map given a general interaction potential. Finally, we provide a renormalization procedure which allows to divide the dependence of the equilibrium map by coupling coefficient from the dependence by the system variables yielding a more rigorous mathematical structure of the nonlinear GLE.