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 deterministic reconstruction of the noise history, using the Mori-Zwanzig projection formalism and the propagation of observables according to the orthogonal dynamics. This reconstruction not only allows to compute the memory kernel but also to investigate other statistical properties of the noise, and to analyze generic projected observables [1,2]. It will be illustrated on a toy model and we will discuss some results in the case of diffusion in a simple fluid, with particular attention to the long-time behaviour and the link with inertia and hydrodynamics . An alternative, stochastic approach to the reconstruction of the noise history will be presented by Hadrien Vroylandt.
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 Molecular hydrodynamics from memory kernels. Lesniki D, Carof A, Vuilleumier R and Rotenberg B. Physical Review Letters, 116 (14), 157804 (2016)