On the estimation of the Mori-Zwanzig memory integral

TL;DR

This paper provides rigorous estimates and convergent approximation methods for the Mori-Zwanzig memory integral, enhancing understanding and computational capability for both linear and nonlinear dynamical systems.

Contribution

The paper introduces mathematically rigorous bounds and convergence conditions for short-memory and hierarchical approximations of the MZ memory integral.

Findings

Derived error bounds and convergence conditions for approximations.

Presented computable upper bounds for the MZ memory integral.

Numerical demonstrations of convergence for linear and nonlinear systems.

Abstract

We develop rigorous estimates and provably convergent approximations for the memory integral in the Mori-Zwanzig (MZ) formulation. The new theory is built upon rigorous mathematical foundations and is presented for both state-space and probability density function space formulations of the MZ equation. In particular, we derive errors bounds and sufficient convergence conditions for short-memory approximations, the $t$-model, and hierarchical (finite-memory) approximations. In addition, we derive computable upper bounds for the MZ memory integral, which allow us to estimate (a priori) the contribution of the MZ memory to the dynamics. Numerical examples demonstrating convergence of the proposed algorithms are presented for linear and nonlinear dynamical systems evolving from random initial states.