Stochastic telegrapher's approach for solving the random Boltzmann-Lorentz gas

Abstract

The 1D random Boltzmann-Lorentz equation has been connected with a set of stochastic hyperbolic equations. Therefore, the study of the Boltzmann-Lorentz gas with disordered scattering centers has been transformed into the analysis of a set of stochastic telegrapher's equations. For global binary disorder (Markovian and non-Markovian) exact analytical results for the second moment, the velocity autocorrelation function, and the self-diffusion coefficient are presented. We have demonstrated that time-fluctuations in the lost of energy in the telegrapher's equation, can delay the entrance to the diffusive regime, this issue has been characterized by a timescale tc which is a function of disorder parameters. Indeed, producing a longer ballistic dynamics in the transport process. In addition, fluctuations of the space probability distribution have been studied, showing that the mean value of a stochastic telegrapher's Fourier mode is a good statistical object to characterize the solution of the random Boltzmann-Lorentz gas. In a different context, the stochastic telegrapher's equation has also been related to the run-and-tumble model in Biophysics. Then a discussion devoted to the potential applications when swimmers' speed and tumbling rate have time fluctuations has been pointed out.