A fractional generalization of the classical lattice dynamics approach

• Main Authors: Michelitsch, T. M, Collet, B. A, Riascos, A. P, Nowakowski, A. F, Nicolleau, F. C. G. A
• Format: Journal Article
• Language: English
• Published: 11-10-2016
• Subjects: Mathematics - Mathematical Physics; Physics - Classical Physics; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.1610.03744

Elsevier, 2016, 92, pp.43-50 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.