THE PARTICLE DIFFUSION IN THE MEDIUM WITH A GIVEN DENSITY OF STATES

Abstract

The particle diffusion in a one-dimensional (1D) lattice as a model of classical localization is studied. This kind of model can describe amorphous bodies. A one-dimensional lattice is a linearly connected system of harmonic oscillators with different  frequencies. This kind of diffusion is described by the density of states of long wavelengths. This kind of diffusion of a given density of states is shown to be abnormal in the case of the dependence on a long time. The general Langevin equation (GLE) can be solved numerically in order to find the second moment of the coordinate x and the correlation function of the velocity for the given initial conditions. The calculation is done by Laplace inverse transformation according to the Talbot algorithm. This model describes the classical 1D (one-dimensional) localization of the particle in the ordered system in space. Here irregularity means that the frequencies  of the oscillations by their values are random according to the given density function of probability. Different diffusion processes are obtained by changing the power dependence  of density of states. It is shown numerically that the correlation function of the velocity of the particle has a negative tail and is described by  time dependence ().