The one-dimensional asymmetric persistent random walk
Résumé
Persistent random walks are intermediate transport processes between a uniform rectilinear motion and a Brownian motion. They are formed by successive steps of random nite lengths and directions travelled at a xed speed. The isotropic and symmetric onedimensional persistent random walk is governed by the telegrapher's equation, also called hyperbolic heat conduction equation. These equations have been designed to resolve the paradox of the in nite speed in the heat and di fusion equations. The niteness of both the speed and the correlation length leads to several classes of random walks: Persistent random walk in one dimension can display anomalies that cannot arise for Brownian motion such as anisotropy and asymmetries. In this work we focus on the case where the mean free path is anisotropic, the only anomaly leading to a physics that is di ferent from the telegrapher's case. We derive exact expression of its Green's function, for its scattering statistics and distribution of rst-passage time at the origin. The phenomenology of the latter shows a transition for quantities like the escape probability and the residence time.
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