Discrete versus continuous -- linear lattice models and their exact continuous counterparts
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.
Cite
@article{arxiv.2601.10184,
title = {Discrete versus continuous -- linear lattice models and their exact continuous counterparts},
author = {Lorenzo Fusi and Oliver Křenek and Vít Průša and Casey Rodriguez and Rebecca Tozzi and Martin Vejvoda},
journal= {arXiv preprint arXiv:2601.10184},
year = {2026}
}
Comments
Extended version of published manuscript