zbMATH — the first resource for mathematics

Velocity and displacement correlation functions for fractional generalized Langevin equations. (English) Zbl 1274.82045
Summary: We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
33E12 Mittag-Leffler functions and generalizations
34A08 Fractional ordinary differential equations
Full Text: DOI
[1] J.-D. Bao, Y.-L. Song, Q. Ji and Y.-Z. Zhuo, Harmonic velocity noise: non-Markovian features of noise-driven systems at long times. Phys. Rev. E 72 (2005), 011113/1-011113/7.
[2] S. Burov and E. Barkai, Fractional Langevin equation: Overdamped, underdamped, and critical behaviors. Phys. Rev. E 78 (2008), 031112/1-031112/18. http://dx.doi.org/10.1103/PhysRevE.78.031112
[3] S. Burov, J.-H. Jeon, R. Metzler and E. Barkai, Single particle tracking in systems showing anomalous diffusion: The role of weak ergodicity breaking. Phys. Chem. Chem. Phys. 13 (2011), 1800-1812. http://dx.doi.org/10.1039/c0cp01879a
[4] S. Burov, R. Metzler and E. Barkai, Aging and nonergodicity beyond the Khinchin theorem. Proc. Natl. Acad. Sci. USA 107 (2010), 13228-13233. http://dx.doi.org/10.1073/pnas.1003693107 · Zbl 1205.82113
[5] R.F. Camargo, A.O. Chiacchio, R. Charnet and E. Capelas de Oliveira, Solution of the fractional Langevin equation and the Mittag-Leffler functions. J. Math. Phys. 50 (2009), 063507/1-063507/8. · Zbl 1216.82028
[6] R.F. Camargo, E. Capelas de Oliveira and J. Vaz Jr, On anomalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator. J. Math. Phys. 50 (2009), 123518/1-123518/13. · Zbl 1373.82056
[7] E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr., Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur. Phys. J., Special Topics 193 (2011), 161-171. http://dx.doi.org/10.1140/epjst/e2011-01388-0
[8] M. Caputo, Elasticità e Dissipazione. Zanichelli, Bologna (1969).
[9] W. Deng and E. Barkai, Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E 79 (2009), 011112/1-011112/7.
[10] M.A. Despósito and A.D. Viñales, Subdiffusive behavior in a trapping potential: Mean square displacement and velocity autocorrelation function. Phys. Rev. E 80 (2009), 021111/1-021111/7. http://dx.doi.org/10.1103/PhysRevE.80.021111
[11] J.L.A. Dubbeldam, V.G. Rostiashvili, A. Milchev and T.A. Vilgis, Fractional Brownian motion approach to polymer translocation: The governing equation of motion. Phys. Rev. E 83 (2011), 011802/1-011802/8. http://dx.doi.org/10.1103/PhysRevE.83.011802 · Zbl 1227.37028
[12] C.H. Eab and S.C. Lim, Fractional generalized Langevin equation approach to single-file diffusion. Physica A 389 (2010) 2510-2521. http://dx.doi.org/10.1016/j.physa.2010.02.041
[13] C.H. Eab and S.C. Lim, Fractional Langevin equations of distributed order. Phys. Rev. E 83 (2011), 031136/1-031136/10. http://dx.doi.org/10.1103/PhysRevE.83.031136
[14] C.H. Eab and S.C. Lim, Accelerating and retarding anomalous diffusion. J. Phys. A: Math. Theor. 45 (2012), 145001/1-145001/17. http://dx.doi.org/10.1088/1751-8113/45/14/145001 · Zbl 1237.35166
[15] K.S. Fa, Generalized Langevin equation with fractional derivative and long-time correlation function. Phys. Rev. E 73 (2006), 061104/1-061104/4. http://dx.doi.org/10.1103/PhysRevE.73.061104
[16] K.S. Fa and J. Fat, Continuous-time random walk: exact solutions for the probability density function and first two moments. Phys. Scr. 84 (2011), 045022/1-045022/6. · Zbl 1267.82054
[17] I. Golding and E.C. Cox, Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96 (2006), 098102/1-098102/4. http://dx.doi.org/10.1103/PhysRevLett.96.098102
[18] R. Gorenflo and F. Mainardi, Random walk models for Space-Fractional Diffusion Processes. Fract. Calc. Appl. Anal. 1, No 2 (1998), 167-192; http://www.math.bas.bg/!fcaa · Zbl 0946.60039
[19] R. Gorenflo and F. Mainardi, Simply and multiply scaled diffusion limits for continuous time random walks. Journal of Physics: Conference Series 7 (2005), 1-16. http://dx.doi.org/10.1088/1742-6596/7/1/001
[20] I. Goychuk, Viscoelastic subdiffusion: From anomalous to normal. Phys. Rev. E 80 (2009), 046125/1-046125/11. http://dx.doi.org/10.1103/PhysRevE.80.046125
[21] Y. He, S. Burov, R. Metzler and E. Barkai, Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett. 101 (2008), 058101/1-058101/4.
[22] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific Publ. Co., Singapore (2000). http://dx.doi.org/10.1142/9789812817747 · Zbl 0998.26002
[23] R. Hilfer, On fractional diffusion and continuous time random walks. Physica A 329 (2003), 35-40. http://dx.doi.org/10.1016/S0378-4371(03)00583-1 · Zbl 1029.60033
[24] J.-H. Jeon and R. Metzler, Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. Phys. Rev. E 81 (2010), 021103/1-021103/11. http://dx.doi.org/10.1103/PhysRevE.81.021103
[25] J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sorensen, L. Oddershede and R. Metzler, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106 (2011), 048103/1-048103/4; http://arxiv.org/abs/1010.0347 http://dx.doi.org/10.1103/PhysRevLett.106.048103
[26] S.C. Kou and X.S. Xie, Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule. Phys. Rev. Lett. 93 (2004), 180603/1-180603/4. http://dx.doi.org/10.1103/PhysRevLett.93.180603
[27] R. Kubo, The fluctuation-dissipation theorem. Rep. Prog. Phys. 29 (1966), 255-284. http://dx.doi.org/10.1088/0034-4885/29/1/306
[28] S.C. Lim and L.P. Teo, Modeling single-file diffusion with step fractional Brownian motion and a generalized fractional Langevin equation. J. Stat. Mech. P08015 (2009).
[29] E. Lutz, Fractional Langevin equation. Phys. Rev. E 64 (2001), 051106/1-051106/4. http://dx.doi.org/10.1103/PhysRevE.64.051106 · Zbl 1308.82050
[30] F. Mainardi and P. Pironi, The fractional Langevin equation: Brownian motion revisited. Extr. Math. 11 (1996), 140-154.
[31] F. Mainardi, Fractional Calculus: Some basic problems in continuum and statistical mechanics. In: A. Carpinteri and F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien and New York (1997), 291-348. · Zbl 0917.73004
[32] R. Mannella, P. Grigolini and B.J. West, A dynamical approach to fractional Brownian motion. Fractals 2 (1994), 81-94. http://dx.doi.org/10.1142/S0218348X94000077
[33] R. Metzler, Generalized Chapman-Kolmogorov equation: A unifying approach to the description of anomalous transport in external fields. Phys. Rev. E 62 (2000), 6233-6245. http://dx.doi.org/10.1103/PhysRevE.62.6233
[34] R. Metzler, E. Barkai and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82 (1999), 3563-3567. http://dx.doi.org/10.1103/PhysRevLett.82.3563
[35] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000), 1-77. http://dx.doi.org/10.1016/S0370-1573(00)00070-3 · Zbl 0984.82032
[36] R. Metzler and J. Klafter, When translocation dynamics becomes anomalous. Biophys. J. 85 (2003), 2776-2779. http://dx.doi.org/10.1016/S0006-3495(03)74699-2
[37] R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004), R161-R208. http://dx.doi.org/10.1088/0305-4470/37/31/R01 · Zbl 1075.82018
[38] I. Podlubny, Fractional Differential Equations. Acad. Press, San Diego etc (1999). · Zbl 0924.34008
[39] N. Pottier, Aging properties of an anomalously diffusing particule. Physica A 317 (2003), 371-382. http://dx.doi.org/10.1016/S0378-4371(02)01361-4 · Zbl 1005.82031
[40] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19 (1971), 7-15. · Zbl 0221.45003
[41] T. Sandev, R. Metzler and Ž. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative. J. Phys. A: Math. Theor. 44 (2011), 255203/1-255203/21. http://dx.doi.org/10.1088/1751-8113/44/25/255203 · Zbl 1301.82042
[42] T. Sandev and Ž. Tomovski, Asymptotic behavior of a harmonic oscillator driven by a generalized Mittag-Leffler noise. Phys. Scr. 82 (2010), 065001/1-065001/4. http://dx.doi.org/10.1088/0031-8949/82/06/065001 · Zbl 1217.82056
[43] T. Sandev, Ž. Tomovski and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise. Physica A 390 (2011), 3627-3636. http://dx.doi.org/10.1016/j.physa.2011.05.039
[44] R.K. Saxena, A.M. Mathai and H.J. Haubold, Unified fractional kinetic equation and a fractional diffusion equation. Astrophysics and Space Sciences 209 (2004), 299-310. http://dx.doi.org/10.1023/B:ASTR.0000032531.46639.a7
[45] R.K. Saxena and M. Saigo, Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function. Fract. Calc. Appl. Anal. 8,No 2 (2005), 141-154; available at shttp://www.math.bas.bg/ fcaa/volume8/fcaa82/saxenasaigo82.pdf. · Zbl 1144.26010
[46] H. Scher H and E.W. Montroll, Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12 (1975), 2455-2477. http://dx.doi.org/10.1103/PhysRevB.12.2455
[47] O.Y. Sliusarenko, V.Y. Gonchar, A.V. Chechkin, I.M. Sokolov, and R. Metzler, Kramers-like escape driven by fractional Gaussian noise. Phys. Rev. E 81 (2010), 041119/1-041119/14. http://dx.doi.org/10.1103/PhysRevE.81.041119
[48] H.M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211 (2009), 198-210. http://dx.doi.org/10.1016/j.amc.2009.01.055 · Zbl 1432.30022
[49] A. Stanislavsky and K. Weron, Numerical scheme for calculating of the fractional two-power relaxation laws in time-domain of measurements. Computer Physics Communications 183 (2012), 320-323. http://dx.doi.org/10.1016/j.cpc.2011.10.014 · Zbl 1268.65031
[50] J. Tang J and R.A. Marcus, Diffusion-controlled electron transfer processes and power-law statistics of fluorescence intermittency of nanoparticles. Phys. Rev. Lett. 95 (2005), 107401/1-107401/4.
[51] Ž. Tomovski, R. Hilfer and H.M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transform. Spec. Func. 21 (2010), 797-814. http://dx.doi.org/10.1080/10652461003675737 · Zbl 1213.26011
[52] Ž. Tomovski, T. Sandev, R. Metzler and J. Dubbeldam, Generalized space-time fractional diffusion equation with composite fractional time derivative. Physica A 391 (2012), 2527-2542. http://dx.doi.org/10.1016/j.physa.2011.12.035
[53] A.D. Viñales and M.A. Despósito, Anomalous diffusion: Exact solution of the generalized Langevin equation for harmonically bounded particle. Phys. Rev. E 73 (2006), 016111/1-016111/4. http://dx.doi.org/10.1103/PhysRevE.73.016111
[54] A.D. Viñales and M.A. Despósito, Anomalous diffusion induced by a Mittag-Leffler correlated noise. Phys. Rev. E 75 (2007), 042102/1-042102/4. http://dx.doi.org/10.1103/PhysRevE.75.042102
[55] A.D. Viñales, K.G. Wang and M.A. Despósito, Anomalous diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise. Phys. Rev. E 80 (2009), 011101/1-011101/6. http://dx.doi.org/10.1103/PhysRevE.80.011101
[56] K.G. Wang and M. Tokuyama, Nonequilibrium statistical description of anomalous diffusion. Physica A 265 (1999), 341-351. http://dx.doi.org/10.1016/S0378-4371(98)00644-X
[57] S.C. Weber, A.J. Spakowitz and J.A. Theriot, Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm. Phys. Rev. Lett. 104 (2010), 238102/1-238102/4.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.