Finite-Velocity Diffusion in Random Media
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Finite-Velocity Diffusion in Random Media Manuel O. Cáceres1,2 Received: 28 January 2020 / Accepted: 11 April 2020 / Published online: 30 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We investigated a diffusion-like equation with a bounded speed of signal propagation (the so called telegrapher’s equation) in a random media. We discuss some properties of the mean-value solution in a well-defined perturbation theory. The frequency-dependent effective-velocity of propagation is studied in the long and short time regime. We show that due to the wave-like character of telegrapher’s equation the effective-velocity is a complex dispersive function in time. Exact results and asymptotic perturbative long-time behaviors (for an exponential space-correlated binary disorder) are presented, showing their agreement and corroborating the goodness of the effective medium approximation in continuous system. Keywords Telegrapher’s equation · Random media · Effective Medium Approximation · Exact solutions Mathematics Subject Classification 82C44 · 82D30 · 60G60 · 60H25 · 58J65
1 Introduction The simplest generalization of a diffusion process, but which is also characterized by a bounded propagation speed is the so-called telegrapher’s equation. Telegrapher’s equation was originally introduced by Lord Kelvin when he was studying dissipation of electromagnetic fields in waveguides. On the other hand, a pioneer work on telegrapher’s equation, from a discrete point of view, was introduced by Golsdtein [1], from which many generalization in the context of persistent random walk were later implemented [2]. In a different context, a comprehensive set of references on heat waves and its connection with telegrapher’s equation can be seen in [3,4], also open questions on thermal waves’ velocity can be read in [5,6]. Telegrapher’s equation, in one dimension, read:
Communicated by Gregory Schehr.
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Manuel O. Cáceres [email protected]
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Comision Nacional de Energia Atomica, Centro Atomico Bariloche and Instituto Balseiro, Universidad Nacional de Cuyo, Av. E. Bustillo 9500, CP 8400 Bariloche, Argentina
2
CONICET, Centro Atomico Bariloche, Av. E. Bustillo 9500, CP 8400 Bariloche, Argentina
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M. O. Cáceres
1 ∂t2 + ∂t − v 2 ∂x2 P (x, t) = 0 T
(1)
where T is a characteristic time and v is a finite propagation speed. The solution of this equation shows at short times (t T ) a wave-like feature, while at long-times the diffusion behavior is restored. Equation (1) shears many applications in diffusion at finite speed [7], Electronics and engineers boundary conditions [8] , Relativistic Brownian motion [9], Persistent random walk [10,11], Anisotropic scattering [12], and finite-velocity diffusion on non-regular lattice [13], to mention a few of them (see also Appendix A). Finite-velocity diffusion models in heterogeneous media is an important problem which, to the best of our knowledge, has not been addressed due to the inherent difficulty in working with continuos disordered syst
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