Simple and efficient algorithms based on Volterra equations to compute memory kernels and projected cross-correlation functions from molecular dynamics

Abstract

Starting from the orthogonal dynamics of any given set of variables with respect to the projection variable used to derive the Mori–Zwanzig equation, a set of coupled Volterra equations is obtained that relate the projected time correlation functions between all the variables of interest. This set of equations can be solved using standard numerical inversion methods for Volterra equations, leading to a very convenient yet efficient strategy to obtain any projected time correlation function or contribution to the memory kernel entering a generalized Langevin equation. Using this strategy, the memory kernel related to the diffusion of tagged particles in a bulk Lennard–Jones fluid is investigated up to the long-term regime to show that the repulsive–attractive cross-contribution to memory effects represents a small but non-zero contribution to the self-diffusion coefficient.

Mobility, response and transport in non-equilibrium coarse-grained models

“…In addition to effective (pair) potentials known from structural coarse-graining [5] the GLE features memory kernels and fluctuating forces to model the frictional interactions and thermal fluctuations in the system. For equilibrium systems, a manifold of different dynamic coarse-graining techniques has been suggested which can be used to systematically derive such GLEs [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], showing the importance and actuality of the topic. The applicability of these methods to nonequilibrium systems is, however, under strong debate.…”

A deep learning approach to the measurement of long-lived memory kernels from generalized Langevin dynamics

“…Many authors in the soft matter and glassy physics communities have taken an alternative approach to determining the memory kernel, whereby an implicit Volterra integral equation for K is constructed from pair-wise correlation functions. 10,[24][25][26][27] Such correlation functions are constructed by taking an ensemble average over a large number of trajectories of particle-based simulations. A vari-ety of different numerical methods have been used to successfully solve such equations for the short-term memory effects of various systems.…”