Discrete versus continuous -- linear lattice models and their exact continuous counterparts
This work addresses a theoretical problem in mathematical physics for researchers studying lattice models and continuum limits, but it is incremental as it reviews and extends existing analysis.
The paper investigates the correspondence between discrete linear lattice models of interacting particles and their continuous counterparts described by linear partial differential equations, focusing on dispersion relations across infinite, periodic, and finite lattices. It frames this as a systematic application of Fourier analysis tools to bridge discrete and continuous settings.
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.