# 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

## References

- Duderstadt, J.J.; Martin, W.R. Transport Theory; J. Wiley: New York, NY, USA, 1979. [Google Scholar]
- Weiss, G.H. Some applications of the persistent random walks and the telegrapher’s equation. Phys. A
**2020**, 311, 381–410. [Google Scholar] [CrossRef] - Shlesinger, M.; Klafter, J.; Zumofen, G. Lévy flights: Chaotic, turbulent and relatisvistic. Fractals
**1995**, 3, 491. [Google Scholar] [CrossRef] - Durian, D.J.; Rudnick, J. Photon migration at short times and distances and in cases of strong absorption. J. Opt. Soc. Am. A
**1997**, 14, 235. [Google Scholar] [CrossRef] - Lemieux, P.A.; Vera, M.U.; Durian, D.J. Diffusing-light spectroscopy beyond the diffusion limit: The role of ballistic transport and anisotropic scattering. Phys. Rev. E
**1998**, 57, 4498. [Google Scholar] [CrossRef] [Green Version] - Durian, D.J.; Rudnick, J. Spatially resolved backscattering: Implementation of extrapolation boundary condition and exponential source. J. Opt. Soc. Am. A
**1999**, 16, 837. [Google Scholar] [CrossRef] - Wang, J.; Dlamini, D.S.; Mishra, S.J.; Pendergast, M.T.M.; Wong, M.C.Y.; Mamba, B.B.; Freger, V.; Verliefde, A.R.D.; Hoek, E.M.V. A critical review of transport through osmotic membranes. J. Membr. Sci.
**2014**, 454, 516–537. [Google Scholar] [CrossRef] - Masoliver, J.; Weiss, G.H. Finite-velocity diffiusion. Eur. J. Phys.
**1996**, 17, 190. [Google Scholar] [CrossRef] - Joseph, D.D.; Preziosi, L. Heat waves. Rev. Mod. Phys.
**1989**, 61, 41. [Google Scholar] [CrossRef] - Jou, D.; Casas-Vázquez, J.; Lebon, G. Extended Irreversible Thermodynamics, 4th ed.; Springer: Berlin, Germany, 2010. [Google Scholar]
- Méndez, V.; Campos, D.; Horsthemke, W. Growth and dispersal with inertia: Hyperbolic reaction-transport systems. Phys. Rev. E
**2014**, 90, 042114. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Goldstein, S. On diffusion by discontinuous movements and on the telegraph equation. Q. J. Mech. Appl. Math.
**1951**, 4, 129. [Google Scholar] [CrossRef] - Kac, M. A stochastic model related to the telegrapher’s equation. Rocky Mt. J. Math.
**1974**, 4, 497. [Google Scholar] [CrossRef] - Masoliver, J.; Lindenberg, K.; Weiss, G.H. A continuous time generalization of the persistent random walk. Phys. A
**1989**, 182, 891. [Google Scholar] [CrossRef] - Masoliver, J.; Lindenberg, K. Continuous-time persistent random walk: A review and some generalizations. Eur. Phys. J. B
**2017**, 90, 107. [Google Scholar] [CrossRef] - Olivares-Robles, M.A.; García-Colín, L.S. Mesoscopic derivation of hyperbolic transport equations. Phys. Rev. E
**1994**, 50, 2451. [Google Scholar] [CrossRef] [PubMed] - Masoliver, J. Random Processes: First-Passage and Escape; World Scientific: Singapore, 2018. [Google Scholar]
- Ishimaru, A.J. Diffusion of light in turbid material. Appl. Opt.
**1989**, 28, 2210. [Google Scholar] [CrossRef] - Heizler, S.I. Asymptotic telegrapher’s equation (P
_{1}) approximation for the transport equation. Nucl. Sci. Eng.**2010**, 166, 17. [Google Scholar] [CrossRef] - Plyukhin, A.V. Stochastic processes leading to wave equations in dimensions higher than one. Phys. Rev. E
**2010**, 81, 021113. [Google Scholar] [CrossRef] [Green Version] - Balescu, R. V-Langevin equations, continuous-time persistent random walks and fractional diffusion. Chaos Solitons Fractals
**2007**, 34, 62. [Google Scholar] [CrossRef] [Green Version] - Weiss, G.H. Aspects and Applications of the Random Walk; North-Holland: Amsterdam, The Netherlands, 1994. [Google Scholar]
- Masoliver, J.; Porrà, J.M.; Weiss, G.H. The continuum limit of a two-dimensional persistent random walk. Phys. A
**1992**, 182, 593. [Google Scholar] [CrossRef] - Porrà, J.M.; Masoliver, J.; Weiss, G.H. A diffusion model incorporating anisotropic properties. Phys. A
**1995**, 218, 229. [Google Scholar] [CrossRef] - Boguñá, M.; Porrà, J.M.; Masoliver, J. Generalization of the persistent random walk to dimensions greater than one. Phys. Rev. E
**1998**, 58, 6992. [Google Scholar] [CrossRef] [Green Version] - Godoy, S.; García-Colín, L.S. Nonvalidity of the telegrapher’s diffusion equation in two and three dimensions for crystalline solids. Phys. Rev. E
**1997**, 55, 2127. [Google Scholar] [CrossRef] - Kolesnik, A.D.; Orsingher, E. A planar random motion with an infinite number of directions controlled by the damped wave equation. J. Appl. Prob.
**2005**, 42, 1168. [Google Scholar] [CrossRef] [Green Version] - Orsingher, E.; De Gregorio, A. Random flights in higher spaces. J. Theor. Prob.
**2007**, 20, 769. [Google Scholar] [CrossRef] - Kolesnik, A.; Pinsky, M.A. Isotropic random motion at finite speed with K-Erlang distributed direction alternatives. J. Stat. Phys.
**2011**, 142, 828. [Google Scholar] - Masoliver, J. Three dimensional telegrapher’s equation and its fractional generalization. Phys. Rev. E
**2017**, 96, 022101. [Google Scholar] [CrossRef] [Green Version] - Masoliver, J.; Lindenberg, K. Two-dimensional telegraphic processes and their fractional generalization. Phys. Rev. E
**2020**, 101, 012137. [Google Scholar] [CrossRef] - Masoliver, J. Fractional telegrapher’s equation from fractional persistent random walks. Phys. Rev. E
**2016**, 93, 052107. [Google Scholar] [CrossRef] [Green Version] - Masoliver, J. Mean first-passage time for non-Markovian continuous noise. Phys. Rev A
**1992**, 45, 2256. [Google Scholar] [CrossRef] - Havlin, S.; Ben-Avraham, D. Diffusion in disorderd media. Adv. Phys.
**1987**, 36, 695. [Google Scholar] [CrossRef] - Bouchaud, J.P.; Georges, A. Anomalous diffusion behavior on disordered media: Statistical mechanics, models and physical applications. Phys. Rep.
**1990**, 195, 127. [Google Scholar] [CrossRef] - Metzler, R.; Klafter, J. The random walk guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2000**, 339, 1–77. [Google Scholar] [CrossRef] - West, B.J.; Bologna, M.; Grigolini, P. Physics of Fractal Operators; Springer: Berlin, Germany, 2003. [Google Scholar]
- Metzler, R.; Klafter, J. The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A
**2004**, 37, R161–R208. [Google Scholar] [CrossRef] - Balescu, R. Aspects of Anomalous Transport in Plasmas; Taylor & Francis: London, UK, 2005. [Google Scholar]
- Eliazar, I.I.; Shlesinger, M.F. Fractional motions. Phys. Rep.
**2013**, 527, 101–129. [Google Scholar] [CrossRef] - West, B.J. Fractional view of complexity: A tutorial. Rev. Mod. Phys.
**2014**, 86, 1169–1184. [Google Scholar] [CrossRef] - West, B.J. Fractional Calculus View of Complexity: Tomorrow’s Science; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Klafter, J.; Sokolov, I. Anomalous diffusion spreads its wings. Phys. World
**2005**, 18, 29. [Google Scholar] [CrossRef] - Montroll, E.W.; Weiss, G.H. Random walks on lattices II. J. Math. Phys.
**1965**, 6, 167–181. [Google Scholar] [CrossRef] - Montroll, E.W.; Shlesinger, M.F. The wonderful world of random walks. In Studies in Statistical Mechanics; Lebowitz, J.L., Montroll, E.W., Eds.; North-Holland: Amsterdam, The Netherlands, 1984; Volume 11. [Google Scholar]
- Kutner, R.; Masoliver, J. The continuous-time random walk still trendy: Fifty-year history, state of the art and outlook. Eur. Phys. J. B
**2017**, 90, 50. [Google Scholar] [CrossRef] [Green Version] - Scher, H.; Montroll, E.W. Random walks on lattices IV. J. Stat. Phys.
**1973**, 9, 101. [Google Scholar] - Scher, H.; Montroll, E.W. Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B
**1975**, 12, 2455. [Google Scholar] [CrossRef] - Ben-Avraham, D.; Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Castiglione, P.; Mazzino, A.; Muratore-Ginanneschi, P.; Vulpiani, A. On strong anomalous diffusion. Phys. D
**1999**, 134, 75. [Google Scholar] [CrossRef] [Green Version] - Gorenflo, R.; Mainardi, F.; Vivoli, A. Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Solitons Fractals
**2007**, 34, 87. [Google Scholar] [CrossRef] [Green Version] - Mainardi, F. The fundamental solution for the fractional diffusion-wave equation. Appl. Math. Lett.
**1996**, 9, 23. [Google Scholar] [CrossRef] [Green Version] - Mainardi, F.; Luchko, Y.; Pagnini, G. The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal.
**2001**, 4, 153. [Google Scholar] - Rebenshtok, A.; Denisov, S.; Hänggi, P.; Barkai, E. Infinite densities for Lévy walks. Phys. Rev. E
**2014**, 90, 062135. [Google Scholar] [CrossRef] [Green Version] - Burshtein, I.; Zharikov, A.A.; Temkin, S.I. Response of a two-level system to a random modulation of the resonance with an arbitrary strong external field. J. Phys. B
**1988**, 21, 1907. [Google Scholar] [CrossRef] - Kofman, A.G.; Zabel, R.; Levine, A.M.; Prior, Y. Non-Markovian stochastic jump processes I. Input field analysis. Phys. Rev. A
**1990**, 41, 6434. [Google Scholar] [CrossRef] [PubMed] - Kingman, J.F.C. Poisson Processes; Oxford University Press: Oxford, UK, 2002. [Google Scholar]
- Roberts, G.E.; Kaufman, H. Table of Laplace Transforms; W. B. Saunders: Philadelphia, PA, USA, 1966. [Google Scholar]
- Claes, I.; Van den Broeck, C. Random walks with persistence. J. Stat. Phys.
**1987**, 49, 383. [Google Scholar] [CrossRef] - Feller, W. An Introduction to Probability Theory and Its Applications; J. Wiley: New York, NY, USA, 1971; Volume II. [Google Scholar]
- Pitt, H.R. Tauberian Theorems; Oxford University Press: Oxford, UK, 1958. [Google Scholar]
- Masoliver, J.; Porrà, J.M.; Weiss, G.H. Solution to the telegrapher’s equation in the presence of reflecting and partly reflecting boundaries. Phys. Rev. E
**1993**, 48, 939. [Google Scholar] [CrossRef] [Green Version] - Magnus, W.; Oberhettinger, F.; Soni, R.P. Formulas and Theorems for the Special Functions of Mathematical Physics; Springer: Berlin, Germany, 1966. [Google Scholar]
- Weiss, G.H. First passage times for correlated random walks and some generalizations. J. Stat. Phys.
**1984**, 37, 325. [Google Scholar] [CrossRef] - Masoliver, J.; Weiss, G.H. First passage times for generalized telegrapher’s equation. Phys. A
**1992**, 183, 537. [Google Scholar] [CrossRef] - Masoliver, J. Telegraphic processes with stochastic resetting. Phys. Rev. E
**2019**, 99, 012121. [Google Scholar] [CrossRef] [Green Version] - Gorenflo, R.; Mainardi, F. Fractional calculus. In Fractals and Fractional Calculus in Continuum Mechanics; Carpinteri, A., Mainardi, F., Eds.; Springer: Berlin, Germany, 1997. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G. Higher Transcendental Functions; McGraw-Hill: New York, NY, USA, 1953; Volume 3. [Google Scholar]
- Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series and Products, 7th ed.; Elsevier: Amsterdam, The Netherlands, 2007. [Google Scholar]
- Handelsman, R.A.; Lew, J.S. Asymptotic expansions of the Laplace convolutions for large argument and fat tail densities for certain sums of random variables. SIAM J. Math. Anal.
**1974**, 5, 425–451. [Google Scholar] [CrossRef] - Mainardi, F.; Pagnini, G. The role of Fox-Wright functions in fractional sub-diffusion of distributed order. J. Comp. Appl. Math.
**2007**, 207, 24. [Google Scholar] [CrossRef] [Green Version] - Mainardi, F. Why the Mittag-Leffler function can be considered the queen function of the fractional calculus? Entropy
**2020**, 22, 1359. [Google Scholar] [CrossRef] - Orsingher, E.; Zhao, X. The space-fractional telegraph equation and the related fractional telegraph process. Chin. Ann. Math.
**2003**, B24, 1. [Google Scholar] [CrossRef] - Orsingher, E.; Beghin, L. Time-fractional telegraph equation and telegraph processes with Brownian time. Probab. Theory Relat. Fields
**2004**, 128, 141. [Google Scholar] - D’Ovidio, M.; Orsingher, E.; Toaldo, B. Time changed processes governed by space-time fractional telegraph equations. Stoch. Anal. Appl.
**2014**, 32, 1009. [Google Scholar] [CrossRef] [Green Version] - Orsingher, E.; Toaldo, B. Space-time fractional equations and the related stable processes at random time. J. Theor. Probab.
**2017**, 30, 1–26. [Google Scholar] [CrossRef] [Green Version] - Lafracte, F.; Orsingher, E. On the fractional wave equation. Mathematics
**2020**, 8, 874. [Google Scholar] [CrossRef]

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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