A fractional generalization of the classical lattice dynamics approach

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 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. The fractional lattice Laplacian matrix contains for α=2 the classical local lattice approach with well known continuum limit of classic local standard elasticity, and for other integer powers to gradient elasticity. We also present a generalization of the fractional Laplacian matrix to n-dimensional cubic periodic (nD tori) and infinite lattices. We show that in the continuum limit the fractional Laplacian matrix yields the well-known kernel of the Riesz fractional Laplacian derivative being the kernel of the fractional power of Laplacian operator. In this way we demonstrate the interlink of the fractional lattice approach with existing continuous fractional calculus. The developed approach appears to be useful to analyze fractional random walks on lattices as well as fractional wave propagation phenomena in lattices.