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Stochastic foundation of normal and anomalous Cattaneo-type transport

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  • Metzler, Ralf
  • Compte, Albert

Abstract

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 find 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 subdiffusion can be found.

Suggested Citation

  • Metzler, Ralf & Compte, Albert, 1999. "Stochastic foundation of normal and anomalous Cattaneo-type transport," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 268(3), pages 454-468.
    • Handle: RePEc:eee:phsmap:v:268:y:1999:i:3:p:454-468
    DOI: 10.1016/S0378-4371(99)00058-8
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    Cited by:

    1. Abel Garcia-Bernabé & S. I. Hernández & L. F. Del Castillo & David Jou, 2016. "Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus," Mathematics, MDPI, vol. 4(4), pages 1-10, December.
    2. Awad, Emad, 2019. "On the time-fractional Cattaneo equation of distributed order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 518(C), pages 210-233.
    3. Thomas M. Michelitsch & Federico Polito & Alejandro P. Riascos, 2023. "Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times," Mathematics, MDPI, vol. 11(2), pages 1-20, January.
    4. Povstenko, Y.Z., 2010. "Evolution of the initial box-signal for time-fractional diffusion–wave equation in a case of different spatial dimensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4696-4707.

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