Fisher and Shannon Functionals for Hyperbolic Diffusion

Abstract

The complexity measure for the distribution in space-time of a finite-velocity diffusion process is calculated. Numerical results are presented for the calculation of Fisher’s information, Shannon’s entropy, and the Cramér–Rao inequality, all of which are associated with a positively normalized solution to the telegrapher’s equation. In the framework of hyperbolic diffusion, the non-local Fisher’s information with the x-parameter is related to the local Fisher’s information with the t-parameter. A perturbation theory is presented to calculate Shannon’s entropy of the telegrapher’s equation at long times, as well as a toy model to describe the system as an attenuated wave in the ballistic regime (short times).