Non-Markovian out-of-equilibrium dynamics: A general numerical procedure to construct time-dependent memory kernels for coarse-grained observables

Abstract

We present a numerical method to compute non-equilibrium memory kernels based on experimental data or molecular dynamics simulations. The procedure uses a recasting of the non-stationary generalized Langevin equation, in which we expand the memory kernel in a series that can be reconstructed iteratively. Each term in the series can be computed based solely on knowledge of the two-time auto-correlation function of the observable of interest. As a proof of principle, we apply the method to crystallization from a super-cooled Lennard Jones melt. We analyze the nucleation and growth dynamics of crystallites and observe that the memory kernel has a time extent that is about one order of magnitude larger than the typical timescale needed for a particle to be attached to the crystallite in the growth regime.

Fluctuation–dissipation relations far from equilibrium: a case study

“…In recent years several different numerical algorithms have been proposed to calculate the memory kernel from microscopic simulations [37,39,44,45]. Here, we employ the most straightforward reconstruction technique, directly based on the numerical inversion of the Volterra equation,…”

Mathematical analysis of some iterative methods for the reconstruction of memory kernels

“…In [11] Meyer, Pelagejcev, and Schilling proposed another iterative method for determining memory kernels in a more general nonequilibrium physical context. Restricted to the equilibrium case (1.1) and rewritten in our notation here, the method employs the very same initial guess (3.1) and proceeds by updating…”

A Numerical Procedure to Evaluate Memory Effects in Non‐Equilibrium Coarse‐Grained Models

“…We have introduced in ref. [26] a method tailored to such a purpose, which needs as a single input the non‐stationary autocorrelation function of the observable under study. We briefly recall the way the method is constructed.…”

“…In addition, the truncation issue reported in ref. [26] is now overcome by turning the problem into the inversion of a matrix without any loss of information. We discuss the computational cost of the method in detail in Appendix C.…”

“…[14][15][16][17][18][19][20][21][22][23][24][25] We have introduced in ref. [26] a method for non-stationary ones. Despite its generality and apparent straightforwardness, this method faces a numerical truncation problem, because the memory kernel is expressed as an infinite series of generalized convolution products that needs to be truncated at some finite order in practice.…”