The generalized Cattaneo (telegrapher's) equation and corresponding random walks (original) (raw)

Stochastic foundation of normal and anomalous Cattaneo-type transport

Physica A: Statistical Mechanics and its Applications, 1999

We investigate the connection of the Cattaneo equation and the stochastic continuous time random walk (CTRW) theory. We show that the velocity model in a CTRW scheme is suited to derive the standard Cattaneo equation, and allows, in principle, for a generalisation to anomalous transport. As a result for a broad waiting time distribution with diverging mean, we ÿnd a strong memory to the initial condition of the system: The ballistic behaviour subsists also for long times. Only if a characteristic waiting time exists, a non-ballistic, enhanced motion is found in the limit of long times. No transition to subdi usion can be found.

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

Physical Review E, 2012

It has been alleged in several papers that the so-called delayed continuous-time random walks (DCTRWs) provide a model for the one-dimensional telegraph equation at microscopic level. This conclusion, being widespread now, is strange, since the telegraph equation describes phenomena with finite propagation speed, while the velocity of the motion of particles in the DCTRWs is infinite. In this paper we investigate the accuracy of the approximations to the DCTRWs provided by the telegraph equation. We show that the diffusion equation, being the correct limit of the DCTRWs, gives better approximations in L 2 norm to the DCTRWs than the telegraph equation. We conclude, therefore, that first, the DCTRWs do not provide any correct microscopic interpretation of the one-dimensional telegraph equation, and second, the kinetic (exact) model of the telegraph equation is different from the model based on the DCTRWs.

The continuous time random walk, still trendy: fifty-year history, state of art and outlook

The European Physical Journal B

In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Phys. J. B Special Issue [http://epjb.epj.org/ open-calls-for-papers/123-epj-b/1090-ctrw-50-years-on] containing articles which show incredible possibilities of the CTRWs.

Relativistic diffusion processes and random walk models

Physical Review D, 2007

The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As well-known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (non-continuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the non-relativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.

A random walk approach to anomalous particle and energy transport

2007

The combined Continuous Time Random Walk (CTRW) in position and momentum space is introduced, in the form of two coupled integral equations that describe the evolution of the probability distribution for finding a particle at a certain position and with a certain momentum as a function of time. The integral equations are solved numerically with a pseudospectral method that is based on the expansion of the unknown functions in terms of Chebyshev polynomials. In parallel, Monte-Carlo simulation are performed. Through the inclusion of momentum space, the combined CTRW is able to yield results on density and temperature profile evolution, on particle and heat fluxes and diffusivities, and on kinetic energy distributions. Depending on the choice of the probability distributions of the particle displacements in position and momentum space, the combined CTRW is able to model phenomena of anomalous transport in position as well as in momentum (or energy or velocity) space. An application is...

Telegrapher’s equations with variable propagation speeds

Physical Review E, 1994

All derivations of the one-dimensional telegrapher's equation, based on the persistent random walk model, assume a constant speed of signal propagation. We generalize here the model to allow for a variable propagation speed and study several limiting cases in detail. We also show the connections of this model with anomalous difFusion behavior and with inertial dichotomous processes.

Universal Spectra of Coherent Atoms in a Recurrent Random Walk

Physical Review Letters, 2009

The probability of a random walker to return to its starting point in dimensions one and two is unity, a theorem first proven by Polya . The recurrence probability -the probability to be found at the origin at a time t, is a power law with a critical exponent −d/2 in dimensions d = 1, 2. We report an experiment that directly measures the Laplace transform of the recurrence probability in one dimension using Electromagnetically Induced Transparency (EIT) of coherent atoms diffusing in a vapor-cell filled with buffer gas. We find a regime where the limiting form of the complex EIT spectrum is universal and only depends on the effective dimensionality in which the random recurrence takes place. In an effective one-dimensional diffusion setting, the measured spectrum exhibits power law dependence over two decades in the frequency domain with a critical exponent of −0.56 ± 0.01. Possible extensions to more elaborate diffusion schemes are briefly discussed.