Lattice fractional Laplacian and its continuum limit kernel on the finite cyclic chain

• Specifying positive physically admissible potentials leading by Hamilton’s variational principle to the fractional Laplacian matrix on the cyclic chain.
• Defining a discrete version of the fractional Laplacian defined on the finite cyclic chain: key operator of newly emerging ‘fractional lattice dynamics’.
• Deriving explicit expressions for the self-adjoint, negative semi-definite fractional Laplacian matrix on infinite and finite cyclic chains.
• Definition of continuum limits for (i) the 1D infinite space and (ii) for the periodic string by employing scaling hypotheses leading to finiteness of elastic energy and total mass of chain.
• Deducing explicit expressions for the fractional Laplacian continuum limit kernels for (i) the infinite space and (ii) the finite periodic string.
• The explicit representations for the cyclic chain fractional Laplacian matrix and its periodic string continuum limit kernel represent key quantities for the description of anomalous transport phenomena such as Lévy flights on finite lattices.

The aim of this paper is to deduce a discrete version of the fractional Laplacian in matrix form defined on the 1D periodic (cyclically closed) linear chain of finite length. We obtain explicit expressions for this fractional Laplacian matrix and deduce also its periodic continuum limit kernel. The continuum limit kernel gives an exact expression for the fractional Laplacian (Riesz fractional derivative) on the finite periodic string. In this approach we introduce two material parameters, the particle mass μ and a frequency Ωα. The requirement of finiteness of the the total mass and total elastic energy in the continuum limit (lattice constant h → 0) leads to scaling relations for the two parameters, namely μ ∼ h   and Ωα2∼h−α. The present approach can be generalized to define lattice fractional calculus on periodic lattices in full analogy to the usual ‘continuous’ fractional calculus.