On the derivation of a Nonlinear Generalized Langevin Equation

Abstract

We recast the Zwanzig's derivation of a non linear generalized Langevin equation (GLE) for a heavy particle interacting with a heat bath in a more general framework showing that it is necessary to readjust the Zwanzig's definitions of the kernel matrix and noise vector in the GLE in order to be able performing consistently the continuum limit. As shown by Zwanzig, the non linear feature of the resulting GLE is due to the non linear 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 non linear GLE.

Memory effects in colloidal motion under confinement and driving

“…the VACF or the MSD), whereas a complementary, bottom-up approach to the GLE is rooted in statistical mechanics and employs the projection operator techniques due to Zwanzig, Mori and others [7][8][9], see [10] for a review. A typical application is the projection on a suspended particle and thereby coarse-graining its interactions with a complex solvent [11,12]; recent progress was made on the non-linear GLE [13][14][15][16] and on GLEs under non-equilibrium conditions [17][18][19][20][21][22]. In essence, the GLE may be seen as a means to encapsulate a large number of unresolved degrees of freedom in the memory function.…”

Non-equilibrium attractor for non-linear stochastic dynamics

“…Systems of interest are usually not isolated but in contact with a much larger environment. What makes their dynamics stochastic is the interaction with the environment (thermal bath): the integration over its degrees of freedom entails that the "force" -understood in a generalised sense-acting on the system becomes effectively random [1,2]. It is in this approach, often called mesoscopic, that the Langevin equation emerges -see ref.…”