Subdiffusion in random compressible flows

In this work, we study the diffusion of admixture particles in a one-dimensional velocity field given by a gradient of a random potential. This refers us to the case of random compressible flows, where previously only scaling estimates were available. We develop a general approach which allows to solve this problem analytically. With its help we derive the macroscopic transport equation and rigorously show in which cases transport can be subdiffusive. We find the Fourier-Laplace transform of the Green's function of this equation and prove that for some potential distributions it satisfies the subdiffusive equation with fractional derivative with respect to time.