Relativistic diffusion processes and random walk models

Dunkel, Jörn; Talkner, Peter; Hänggi, Peter

doi:10.1103/physrevd.75.043001

Cited by 65 publications

(75 citation statements)

References 51 publications

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“…27,38 An appealing feature of that hyperbolic equation is that its solutions are characterized by a finite signal propagation speed. Unfortunately, the solutions typically contain singular, d-functional components or sharp peaks at propagating diffusion fronts.…”

Section: Discussionmentioning

confidence: 99%

“…16,17 The hyperbolic nature of the telegraph equation denotes that the telegraph approximation typically yields solutions that contain singular components or sharp spikes at propagating diffusion fronts. 27 Although the d-functional singularities of the analytical solutions cannot be resolved numerically, the solutions of boundary value problems are also predicted to contain discontinuities that are absent in the solutions of the underlying Fokker-Planck equation. 28,29 Malkov 30 and Malkov and Sagdeev 26 recently reiterated these unsatisfactory features of the telegraph approximation and advocated the use of an alternative-the hyperdiffusion approximation.…”

Section: Introductionmentioning

confidence: 99%

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Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport

Litvinenko

Noble

2016

Physics of Plasmas

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The telegraph equation and its generalizations have been repeatedly considered in the models of diffusive cosmic-ray transport. Yet the telegraph model has well-known limitations, and analytical arguments suggest that a hyperdiffusion model should serve as a more accurate alternative to the telegraph model, especially on the timescale of a few scattering times. We present a detailed sideby-side comparison of an evolving particle density profile, predicted by the telegraph and hyperdiffusion models in the context of a simple but physically meaningful initial-value problem, compare the predictions with the solution based on the Fokker-Planck equation, and discuss the applicability of the telegraph and hyperdiffusion approximations to the description of strongly anisotropic particle distributions. Published by AIP Publishing. [http://dx

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport

Litvinenko

Noble

2016

Physics of Plasmas

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“…In the context of diffusion theory, the telegrapher's equation (TE) is seen as a relativistic generalization of the diffusion equation, since the latter is not compatible with relativity [140][141][142][143][144]. It also takes into account ballistic motion, and tends to be more accurate in modeling transport near boundaries than the diffusion equation [145].…”

Section: Fractional Telegrapher's Equationmentioning

confidence: 99%

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

2017

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Abstract.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. Inspiring properties and achievementsIn their pioneering work published in year 1965 [1], physicists Eliott W. Montroll and George H. Weiss introduced the concept of continuous-time random walk (CTRW) as a way to make the interevent-time continuous and fluctuating. It is characterized by some distribution associated with a stochastic process, giving an insight into the process activity. This distribution, called pausing-or waiting-time one (WTD), permitted the description of both Debye (exponential) and, what is most significant, non-Debye (slowly-decaying) relaxations as well as normal and anomalous transport and diffusion [2,3] -thus the model involves fundamental aspects of the stochastic world -a real, complex world. Notably, ancestors of this concept are presented by Michael Shlesinger in [4].Let us incidentally comment that term "walk" in the name "continuous-time random walk" is commonly used in the generic sense comprising two concepts: namely, both the walk (associated with finite displacement velocity of the process) and flight (associated with an instantaneous displacement of the process). Thus we have to specify in a detailed way with what kind of process we are considering.The CTRW formalism was most conveniently developed by physicists Scher and Lax in terms of recursion relations [5][6][7][8][9][10]. In this context the distinction between Contribution to the Topical Issue "Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver. a e-mail: ryszard.kutner@fuw.edu.pl discrete and continuous times [11] and also between separable and non-separable WTDs were introduced [12]. A thorough analysis of the latter, called also the nonindependent CTRW, was performed a decade ago [13] although, in the context of the concentrated lattice gases, it was performed much earlier [14,15]. These analyses took into account dependences over many correlated consecutive particle displacements and waiting (or interevent) times. The Scher and Lax formulation of the CTRW formalism is particularly convenient to study as well anomalous transport and diffusion as the non-Debye relaxation and their anomalous scaling properties (e.g., the nonlinear time growth of the process variance). Examples are the span of the walk, the first-passage times, survival probabilities, the number of distinct sites visited and, of course, mean and mean-square displacement if they exist. It is very interesting...

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“…For times T > 0, the asymptotic behavior of the intensity profile lim z→∞Ī (z) = 0 includes a nonvanishing probability of particles travelling faster than the speed of light. A non-Markovian generalization of this diffusion model, as described in Dunkel et al (2007), could avoid this violation.…”

Section: Intensity Profilesmentioning

confidence: 99%

A diffusive description of the focused transport of solar energetic particles

Artmann

Schlickeiser

Agueda

et al. 2011

A&A

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The transport of solar energetic charged particles along the interplanetary magnetic field in the ecliptic plane of the sun can be described roughly by a one-dimensional diffusion equation. Large-scale spatial variations of the guide magnetic field can be taken into account by adding an additional term to the diffusion equation that includes the effect of adiabatic focusing. We solve this equation analytically by assuming a point-like particle injection in time and space and a spatial power-law dependence for the focusing length and the spatial diffusion coefficient. We infer the intensity-and anisotropy-time profiles of solar energetic particles from this solution. Through these the influence of different assumptions for the diffusion parameters can be seen in a mathematically closed form. The comparison of calculated and measured intensity-and anisotropy-time profiles, which are a powerful diagnostic tool for interplanetary particle transport, gives information about the large-scale spatial dependence of the focusing length and the diffusion coefficient. For an exceptionally large solar energetic particle event, which did occur on 2001 April 15, we fit the 27−512 keV electron intensities and anisotropies observed by the Wind spacecraft using the theoretically derived profiles. We find a linear spatial dependence of the mean free path along the guiding magnetic field. We also find the mean free path to be energy independent, which supports the theory of "velocity-dependent diffusion". This means that the intensity profiles for the discussed energies exhibit the same shape if they are plotted against the traveled distance and not against the time. In this case the profiles differ only in their maximum values and we can determine the energy spectra of the solar flare electrons out of the scaling factor we need to fit the data. The derived spectra exhibits a power-law dependence ∝E −3 kin in an energy range from ∼50 keV to ∼500 keV.

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Relativistic diffusion processes and random walk models

Cited by 65 publications

References 51 publications

Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport

Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport

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

A diffusive description of the focused transport of solar energetic particles

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