The derivation of Generalized Langevin Equations for a set of coarse-grained variables requires the ability to compute the memory kernel from the underlying dynamics of the full system.
Various strategies have been proposed to achieve this goal, such as solving the integro-differential equation satisfied by the kernel or optimizing a (parametric or time-discretized) kernel to best reproduce
time correlation functions.
Here, we will discuss an approach based on the probabilistic reconstruction of the noise history for the coarse-grained variables of interest, using extended dynamics with hidden variables.
Such extended dynamics have been promoted for the integration of Generalized Langevin Equations, by reintroducing additional degrees of freedom following Markovian dynamics.
We propose to reconstruct the dynamics of the hidden variables from the trajectory of the ones of interest (observed on the initial, non-coarse-grained system), following a maximum likelihood approach. The subsequent analysis of the extended dynamics then provides statistical information on the noise history, including the corresponding memory kernel.
The method will be illustrated on several model systems, both in and out of equilibrium, including a simulation of Lennard-Jones particles.
An alternative, deterministic approach to the reconstruction of the noise history will be presented by Rodolphe Vuilleumier and Benjamin Rotenberg.