On the connection between biased dichotomous diffusion and the one-dimensional Dirac equation

Abstract

The master equation for dichotomous diffusion (DD) (the integral of a random telegraph process) is the well-known telegrapher's equation, which is converted to the Klein–Gordon equation by a simple transformation. After a brief recapitulation of the solution and of the analogy between DD and the Dirac equation in one spatial dimension, we consider velocity-biased DD. The corresponding master equation and its solution are presented. It is shown that these may be interpreted physically in terms of a Lorentz transformation to a frame moving with a boost velocity equal to the mean drift velocity of the diffusing particle. The modifications that arise in the connection with the Dirac equation are also exhibited. The correspondence between the rest mass of the Dirac particle and the frequency of direction reversal in the DD is shown to be modified precisely by the time dilatation correction to the latter quantity.