Long-range interactions and nonextensivity in one-dimensional systems

Abstract

The effect of long-range interactions and its connection with nonextensivity was analyzed by determining the propagating properties of wave packets in a one-dimensional ordered structure. It was assumed that a power-law decaying hopping term with an exponent $\ensuremath{\alpha}$ determined the range of the interaction. One can clearly notice different regimes of propagation according to the different $\ensuremath{\alpha}$ values. When $\ensuremath{\alpha}=0,$ a situation in which every site in the lattice sees the others with the same intensity, the wave packet gets localized around the starting position, i.e., self-trapping takes place. By increasing $\ensuremath{\alpha}$ the localization is lost, the ${X}_{\mathrm{MSD}}\ensuremath{\equiv}〈{n}^{2}〉$ grows with time but shows oscillations that disappear as soon as $\ensuremath{\alpha}$ reaches the value $1.$ In this case, and for very short times $(t\ensuremath{\lesssim}2 \mathrm{ps}),$ the packet diffuses with a diffusion coefficient that increases with the number of sites in the lattice, that is $〈{n}^{2}〉{=D}_{1}(N)t.$ This size effect reported here is absent in the familiar nearest-neighbor (NN) interaction case. For later times and $\ensuremath{\alpha}=1,$ the particle propagates subdiffusively. As for bigger $\ensuremath{\alpha}$ values, the propagation can be characterized by the following ${X}_{\mathrm{MSD}}{=Ct}^{\ensuremath{\mu}}$ where the exponent $\ensuremath{\mu}$ approaches rapidly the value $2$ as long as $\ensuremath{\alpha}$ is greater than $2.$ In other words, for great $\ensuremath{\alpha}$'s we have ballistic propagation, as was the case for the NN interaction. When a dc electric field is applied we get a localized wave packet, i.e., it is the phenomenom of dynamic localization.