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

Amaël Obliger¹

¹Institut des Sciences Moléculaires, University of Bordeaux-Bordeaux INP - CNRS, UMR 5255, F-33400 Talence, France.

The Journal of Chemical Physics

|April 15, 2023

Summary

A new method solves Volterra equations to calculate projected time correlation functions and memory kernels for generalized Langevin equations. This analysis of tagged particle diffusion in a Lennard-Jones fluid reveals a small, non-zero cross-contribution to self-diffusion.

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Area of Science:

Statistical Mechanics
Computational Physics

Background:

The Mori-Zwanzig formalism is a key tool for deriving generalized Langevin equations (GLEs) from microscopic dynamics.
Calculating memory kernels and projected time correlation functions within this framework can be computationally intensive.

Purpose of the Study:

To develop an efficient and convenient strategy for obtaining projected time correlation functions and memory kernel contributions for GLEs.
To investigate the role of repulsive-attractive cross-contributions to memory effects in the self-diffusion of tagged particles.

Main Methods:

Derivation of coupled Volterra equations from orthogonal dynamics of system variables.
Application of standard numerical inversion methods to solve the obtained Volterra equations.
Investigation of the memory kernel for tagged particle diffusion in a bulk Lennard-Jones fluid.

Main Results:

A set of coupled Volterra equations was derived, relating projected time correlation functions.
A computationally efficient strategy was established for calculating projected time correlation functions and memory kernel contributions.
The repulsive-attractive cross-contribution to memory effects was found to be small but non-zero for self-diffusion.

Conclusions:

The developed method provides a convenient and efficient approach for analyzing memory effects in complex systems.
The study quantifies the contribution of specific interactions to the self-diffusion coefficient in a Lennard-Jones fluid.