On the Asymptotic Behavior of the Kernel Function in the Generalized Langevin Equation: A One-dimensional lattice model

TL;DR

This paper analyzes the decay properties of the memory kernel in a one-dimensional lattice model's generalized Langevin equation, revealing power-law behaviors and dependence on coarse-graining levels.

Contribution

It provides explicit matrix expressions and estimates for the memory kernel, advancing understanding of its asymptotic behavior in lattice models.

Findings

Kernel exhibits power-law decay spatially and temporally

Explicit matrix form of the kernel derived

Decay properties depend on coarse-graining level

Abstract

We present some estimates for the memory kernel function in the generalized Langevin equation, derived using the Mori-Zwanzig formalism from a one-dimensional lattice model, in which the particles interactions are through nearest and second nearest neighbors. The kernel function can be explicitly expressed in a matrix form. The analysis focuses on the decay properties, both spatially and temporally, revealing a power-law behavior in both cases. The dependence on the level of coarse-graining is also studied.