Abstract
We develop a thorough mathematical analysis to deduce conditions for the accuracy and convergence of different approximations of the memory integral in the Mori-Zwanzig (MZ) equation. In particular, we derive error bounds and sufficient convergence conditions for short-memory approximations, the t-model, and hierarchical (finite-memory) approximations. In addition, we derive useful upper bounds for the MZ memory integral, which allow us to estimate a priori the contribution of the MZ memory to the dynamics. Such upper bounds are easily computable for systems with finite-rank projections. Numerical examples are presented and discussed for linear and nonlinear dynamical systems evolving from random initial states.
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