Finite-velocity diffusion

Abstract

The phenomenon of diffusion modelled as Brownian motion or in terms of the diffusion equation is deficient in that it permits an infinite speed of signal propagation. The simplest diffusion-like equation with a bounded speed of signal propagation is the so-called telegrapher's equation which implicitly contains a type of momentum. We discuss some properties, solutions and problems related to this equation. These are generally more complicated in form than corresponding ones for the diffusion equation, exhibiting wave-like properties at short times and diffusion-like properties in the long-time limit. Resumen. El fenómeno de la difusión, modelado tanto como movimiento Browniano como mediante la ecuación de difusión, es deficiente en el sentido que permite una velocidad de propagación infinita. La denominada ecuación del telegrafista es una de las ecuaciones más simples con una velocidad de propagación acotada y que contiene de forma implícita un cierto tipo momento. En este trabajo se discuten algunas propiedades, soluciones y cuestiones relacionadas con esta ecuación. Estas propiedades y soluciones son generalmente más complicadas que las correspondientes a la ecuación de difusión ordinaria, presentando, a tiempos cortos, propiedades de tipo ondulatorio y propiedades difusivas en el límite asintótico.

Delayed uncoupled continuous-time random walks do not provide a model for the telegraph equation

“…Thus, the jump model may, in contrast to the kinetic * rukol@ammp.ioffe.ru † samsonov@math.ioffe.ru one, violate the principle of causality. This is similar to the classic diffusion paradox [19][20][21][22][23] related to the conventional diffusion equation, which is widely used for approximate macroscopic description of nonanomalous diffusion and Brownian motion. The difference is that in classic Einstein's model of Brownian motion [24], leading to the diffusion equation, the particle moves with finite velocity.…”

“…The telegraph equation [19,20,22,23,26] was proposed as an alternative to the diffusion equation. In contrast to the latter, which is parabolic, the telegraph equation is hyperbolic, providing the finite speed of signal propagation.…”

Relativistic diffusion processes and random walk models

“…Several different relativistic generalizations of the nonrelativistic diffusion equation (1) have been proposed in the past [8,10,11,12,13,14,15,16]. The most prominent example is the telegraph equation (3).…”

“…This problem has attracted considerable interest during the past 100 years (see, e.g., [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and references therein). Nonetheless, it seems fair to say that a commonly accepted solution is still outstanding.…”

“…In this context, an often considered alternative to Eq. (1) is given by the telegraph equation (TE) [5,10,13,14,15,16,22,30,31,32] …”