Nonlocal-in-time finite-velocity diffusion approach on a ring

The ubiquitous telegrapher's equation is presented in the context of a non-local-in-time master equation on the lattice. From the exact solution of this transport equation, for different hopping models, the second moment in the infinite lattice and the time evolution of the probability in the ring have been analyzed as a function of the two characteristic timescales appearing in the memory kernel of the finite-velocity approach: the rate of energy loss and the timescale characterizing the jumping process in the lattice. We have demonstrated how these timescales characterize the constraint to find positive solutions, the time variation of entropy and therefore the approach to the disordered stationary state on the ring. This lattice model provides an analytic treatment. Thus, this result is relevant in the study of Shannon entropy, transport of information, and waves in lattices and sheds light on the functional role of the loss of energy in the finite-velocity diffusion dynamics.