Abstract

In many cases anomalous diffusion processesexhibit a bi-fractal property. For example  active transport in the cell breaks the basic concepts of scaling.In the presence of ATP , $q>0$ moments of the tracer particle displacement$\langle |x|^q \rangle$, increase eithersuper-diffusively or quasi  ballistically for values of $q$ below or abovea critical value $q_c$ respectively. This dual nature of the transport isfound in  many systems including deterministic modelslike  the Lorentz gas [1]  and stochastic approaches like the L\’evy walk. Given thatstandard fractional  diffusion equationsfail to describe this widely observed  bi-scalingwe investigate the problem using the widely applicable L\’evy walk model.  We showthat non-normalised infinite densities are complementary to the standard L’evy-Gauss  central limittheorems in the statistical description of the process [2.3]. [1] Lior Zarfaty,Alexander Peletskyi,EB, andSergey DenisovInfinite horizon billiards: Transport at the border between Gauss and L\’evy universality classes, Phys. Rev. E. 100, 042140 (2019).   [2] A. Rebenshtok, S. Denisov, P. H\”anggi, and E. Barkai,Non-normalizable densities in strong anomalous diffusion: beyondthe central limit theorem, Phys. Rev. Letters, 112, 110601 (2014).  [3] A. Rebenshtok, S. Denisov, P. H\”anggi, and E. Barkai, Infinite densities for L\’evy walks Phys. Rev. E.90, 062135 (2014).