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V-Langevin equations, continuous time random walks and fractional diffusion

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  • Balescu, R.

Abstract

The following question is addressed: under what conditions can a strange diffusive process, defined by a semi-dynamical V-Langevin equation or its associated hybrid kinetic equation (HKE), be described by an equivalent purely stochastic process, defined by a continuous time random walk (CTRW) or by a fractional differential equation (FDE)? More specifically, does there exist a class of V-Langevin equations with long-range (algebraic) velocity temporal correlation, that leads to a time-fractional superdiffusive process? The answer is always affirmative in one dimension. It is always negative in two dimensions: any algebraically decaying temporal velocity correlation (with a Gaussian spatial correlation) produces a normal diffusive process. General conditions relating the diffusive nature of the process to the temporal exponent of the Lagrangian velocity correlation (in Corrsin approximation) are derived. It is shown that a bifurcation occurs as the latter parameter is varied. Above that bifurcation value the process is always diffusive.

Suggested Citation

  • Balescu, R., 2007. "V-Langevin equations, continuous time random walks and fractional diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 62-80.
    • Handle: RePEc:eee:chsofr:v:34:y:2007:i:1:p:62-80
    DOI: 10.1016/j.chaos.2007.01.050
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    Cited by:

    1. Liu, Q.X. & Liu, J.K. & Chen, Y.M., 2017. "An analytical criterion for jump phenomena in fractional Duffing oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 216-219.
    2. Pagnini, Gianni, 2014. "Short note on the emergence of fractional kinetics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 409(C), pages 29-34.
    3. Villarroel, Javier & Montero, Miquel, 2009. "On properties of continuous-time random walks with non-Poissonian jump-times," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 128-137.
    4. Lin, Guoxing, 2018. "General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 497(C), pages 86-100.
    5. Guoxing Lin, 2018. "Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches," Mathematics, MDPI, vol. 6(2), pages 1-16, January.
    6. Zheng, Yunying & Zhao, Zhengang & Cui, Yanfen, 2019. "The discontinuous Galerkin finite element approximation of the multi-order fractional initial problems," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 257-269.
    7. Richard L. Magin & Ervin K. Lenzi, 2021. "Slices of the Anomalous Phase Cube Depict Regions of Sub- and Super-Diffusion in the Fractional Diffusion Equation," Mathematics, MDPI, vol. 9(13), pages 1-29, June.
    8. Fan Yang & Qu Pu & Xiao-Xiao Li & Dun-Gang Li, 2019. "The Truncation Regularization Method for Identifying the Initial Value on Non-Homogeneous Time-Fractional Diffusion-Wave Equations," Mathematics, MDPI, vol. 7(11), pages 1-21, October.

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