# Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions

## Abstract

**:**

## 1. Introduction

_{1}approximation to the full transport equation for which the basic assumption is that the change in the direction of motion due to a single scattering event is small [1,2,18,19]. In a more recent approach [20] a three-dimensional TE model is obtained by a modification of the continuity equation for the probability current. The model is, however, limited to a discrete number of transport directions, which restricts possible applications. Other approaches suppose phenomenological generalizations, where a three dimensional TE is postulated for uniform isotropic media by assuming the same form as the one-dimensional TE, but with numerical corrections in the coefficients which guarantee correct ballistic ($t\to 0$) and diffusive ($t\to \infty $) behaviors in three dimensions [4,5,6]. The more fundamental and less phenomenological way of describing telegraphic processes is, however, based on random walk models since they try to reproduce the microscopic mechanism of transport.

## 2. Continuous Multistate Random Walk in Three Dimensions

#### 2.1. General Setting

#### 2.2. Independent Scattering

#### 2.3. The Isotropic and Uniform Random Walk

## 3. Telegrapher’s Equation

#### 3.1. Fluid Limit Approximation

#### 3.2. The Three-Dimensional Telegrapher’s Equation

## 4. The Two Dimensional Case

#### 4.1. General Model

#### 4.2. The Isotropic and Uniform Case

#### 4.3. Fluid Limit Approximation and Telegrapher’s Equation

## 5. Fractional Transport

#### 5.1. The Fractional Isotropic Walk

#### 5.2. Fractional Telegrapher’s Equation in Three Dimensions

#### 5.3. Lower Dimensional Cases

#### 5.3.1. One Dimension

#### 5.3.2. Two Dimensions

#### 5.4. Characteristic Function

## 6. Time-Fractional Telegraphic Transport

#### 6.1. Laplace Transform of the PDF

#### 6.1.1. One Dimension

#### 6.1.2. Two Dimensions

#### 6.1.3. Three Dimensions

#### 6.2. Long-Time Asymptotic Expressions

#### 6.2.1. One Dimension

#### 6.2.2. Two Dimensions

#### 6.2.3. Three Dimensions

#### 6.3. Moments of the Effective Distance Travelled

#### 6.3.1. One Dimension

#### 6.3.2. Two Dimensions

#### 6.3.3. Three Dimensions

## 7. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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Masoliver, J.
Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions. *Entropy* **2021**, *23*, 364.
https://doi.org/10.3390/e23030364

**AMA Style**

Masoliver J.
Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions. *Entropy*. 2021; 23(3):364.
https://doi.org/10.3390/e23030364

**Chicago/Turabian Style**

Masoliver, Jaume.
2021. "Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions" *Entropy* 23, no. 3: 364.
https://doi.org/10.3390/e23030364