Complementary Densities of Levy Walks: Typical and Rare Fluctuations

Strong anomalous diffusion is a recurring phenomenon in many fields, ranging from the spreading of cold atoms in optical lattices to transport processes in living cells. For such processes the scaling of the filaments follows <vertical bar x(t)vertical bar(q)> similar to t(qv(q)) and is characterized by a bi-linear spectrum of the scaling exponents, qv(q). Here we analyze Levy walks, with power law distributed times of flight psi(tau) similar to tau(-(1+alpha)) demonstrating sharp bi-linear scaling. Previously We showed that for alpha > 1 the asymptotic behavior is characterized by two complementary densities corresponding to the hi scalingof the moments of (t). The first density is the expected generalized central limit theorem which is responsible for the low-order moments 0 < q < alpha. The second one, a non-normalizable density (also called infinite density) is formed by rare fluctuations and determines the time evolution of higher-order moments. Here we use the Fah di Bruno formalism to derive the moments of sub-ballistic super-diffusive Levy walks and then apply the Mellin transform technique to derive exact expressions for their infinite densities. We find a uniform approximation for the density of particles using Levy distribution for typical fluctuations and the infinite density for the rare ones. For ballistic Levy walks 0 < alpha < 1 we obtain mono-scaling behavior which is quantified.