Fractional diffusion, irreversibility and entropy

Three types of equations linking the diffusion equation and the wave equation are studied: the time fractional diffusion equation, the space fractional diffusion equation and the telegraphers equation. For each type, the entropy production is calculated and compared. It is found that the two fractional diffusions, considered as linking bridges between reversible and irreversible processes, possess counterintuitive properties: as the equation becomes more reversible, the entropy production increases. The telegraphers equation does not have the same counterintuitive behavior. It is suggested that the different behaviors of these equations might be related to the velocities of the corresponding random walkers.