A fractional generalization of the classical lattice dynamics approach

TL;DR

This paper introduces a fractional lattice model based on Hamilton's principle that generalizes classical difference operators to fractional powers, capturing nonlocal effects and describing anomalous transport phenomena in discrete and continuous systems.

Contribution

It develops a physically admissible fractional lattice model using Hamilton's variational principle, extending difference operators to fractional powers and connecting discrete models with continuous fractional Laplacians.

Findings

Explicit fractional Laplacian matrices for 1D chains derived

Model captures nonlocal long-range interactions in lattice systems

Continuum limit yields Riesz fractional derivative kernels

Abstract

We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and infinite lattice in n=1,2,3,..n=1,2,3,.. dimensions. The present model which is based on Hamilton's variational principle is confined to conservative non-dissipative isolated systems. The present approach yields the discrete analogue of the continuous space fractional Laplacian kernel. As continuous fractional calculus generalizes differential operators such as the Laplacian to non-integer powers of Laplacian operators, the fractional lattice approach developed in this paper generalized difference operators such as second difference operators to their fractional (non-integer) powers. Whereas differential operators and difference operators constitute local operations, their fractional generalizations introduce nonlocal long-range features. This is true for discrete and continuous fractional operators. The nonlocality property of the lattice fractional Laplacian matrix allows to describe numerous anomalous transport phenomena such as anomalous fractional diffusion and random walks on lattices. We deduce explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear chains and their Riesz fractional derivative continuum limit kernels.