A new theory for anomalous diffusion with a bimodal flux distribution

This paper deals with a new governing equation for anomalous diffusion encompassing a large spectrum of phenomena with particular attention on delaying processes. The analysis starts with a discrete approach and a law of evolution introducing a partial retention of the diffusing particles at each time step. The resulting differential equation assuming that the concentration function belongs to class C3 is a fourth-order differential equation. To fit this result into the framework of a new theory, a bi-modal flux distribution for the diffusion process associated with two energy states is proposed. The first energy state is related to the set of particles flowing according to the Fick’s law and the complementary set follows a new law. Two key parameters are introduced, namely, a parameter β indicating the fraction of the particles in the principal energy state and a parameter R controlling the effect of the secondary flux. Some examples are presented characterizing different types of phenomena as function of the relative values of β and R. The necessary conditions for the retention behavior are discussed for some particular cases.