Scaling forms of particle densities for Lévy walks and strong anomalous diffusion

We study the scaling behavior of particle densities for Levy walks whose transition length r is coupled with the transition time t as vertical bar r vertical bar alpha t(alpha) with an exponent alpha > 0. The transition-time distribution behaves as psi(t) alpha t(-1-beta) with beta > 0. For 1 < beta < 2 alpha and alpha >= 1, particle displacements are characterized by a finite transition time and confinement to vertical bar r vertical bar < ta while the marginal distribution of transition lengths is heavy tailed. These characteristics give rise to the existence of two scaling forms for the particle density, one that is valid at particle displacements vertical bar r vertical bar << t(alpha) and one at vertical bar r vertical bar less than or similar to t(alpha). As a consequence, the Levy walk displays strong anomalous diffusion in the sense that the average absolute moments <vertical bar r vertical bar(q)> scale as t(q nu(q)) with nu(q) piecewise linear above and below a critical value q(c). We derive explicit expressions for the scaling forms of the particle densities and determine the scaling of the average absolute moments. We find that <vertical bar r vertical bar > alpha t(q alpha/beta) for q < q(c) = beta/alpha and <vertical bar r vertical bar(q)> alpha t(1+alpha q-beta) for q > q(c). These results give insight into the possible origins of strong anomalous diffusion and anomalous behaviors in disordered systems in general