Nonextensive effects in tight-binding systems with long-range hopping

Consequences of long-range hopping in one-dimensional tight-binding models are studied. A hopping term proportional to 1/r(ij)(alpha) is used, where r (ij) denotes the distance between atoms i and j and alpha determines the ra nge of the interactions within the system. Calculations of the diffusion of an electron along the lattice yield interesting effects of nonextensivity, In particular, we find that the mean square displacement scales anomalousl y as Dt(gamma) in the following way: For 0 < alpha < 1, we find D proportio nal to NN*, where N is the number of atoms on the lattice and N* = N1-alpha -1/1-alpha is related to the number of elements interacting at a given alph a. In this regime the behaviour is subdiffusive (.5 less than or equal to g amma < 1) but approaches normal diffusion (gamma = 1) for alpha = i. There exists a transition region between 1 < alpha < 2, where the diffusion coeff icient loses its system size dependency and becomes size independent for al l alpha greater than or equal to 2. In addition, we find 1 < gamma less tha n or equal to 2 (superdiffusion) for alpha > i. Ballistic motion (gamma = 2 ) is recovered for all alpha greater than or equal to 1.5 and is maintained in the nearest neighbour limit. Specific heat and internal energy as a fun ction of temperature and system size are also analyzed. They appear extensi ve on the macroscopic level for all values of alpha.