Discrete versus continuous -- linear lattice models and their exact continuous counterparts

TL;DR

This paper systematically explores the relationship between discrete lattice models of interacting particles and their continuous PDE counterparts, focusing on Fourier analysis and dispersion relations across various boundary conditions.

Contribution

It provides a comprehensive analysis of the correspondence between discrete and continuous linear models, extending from infinite to finite lattices with different boundary conditions.

Findings

Established the Fourier analysis framework for model correspondence.

Analyzed dispersion relations for various lattice boundary conditions.

Clarified the transition from discrete to continuous models in linear systems.

Abstract

We review and study the correspondence between discrete linear lattice/chain models of interacting particles and their continuous counterparts represented by linear partial differential equations. In particular, we study the correspondence problem for linear nearest neighbour interaction lattice models as well as for linear multiple-neighbour interaction lattice models, while we gradually proceed from infinite lattices to periodic lattices and finally to finite lattices with fixed ends/zero Dirichlet boundary conditions. The whole study is framed as a systematic specialisation of Fourier analysis tools from the continuous to the discrete setting and vice versa, and the correspondence between the discrete and continuous models is examined primarily with regard to the dispersion relation.