Free Access
Math. Model. Nat. Phenom.
Volume 8, Number 2, 2013
Anomalous diffusion
Page(s) 127 - 143
Published online 24 April 2013
  1. A. Taloni, A. Chechkin, J. Klafter. Generalized elastic model yields a fractional Langevin equation description. Phys. Rev. Lett. 104 (2010), No 16, 160602-1-4. [CrossRef] [PubMed] [Google Scholar]
  2. A. Saichev, M. Zazlawsky. Fractional kinetic equations: solutions and applications. Chaos, 7 (1997), No 4, 753-765. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  3. S. G. Samko, A. A. Kilbas, O. I. Marichev. Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Amsterdam, 1993. [Google Scholar]
  4. M. Doi, S. F. Edwards. The Theory of Polymer Dynamics. Clarendon, Oxford, 1986. [Google Scholar]
  5. R. Granek. From semi-flexible polymers to membranes: anomalous diffusion and reptation. J. Phys. II France, 7 (1997), 1761-1788. [CrossRef] [EDP Sciences] [Google Scholar]
  6. E. Farge, A. C. Maggs. Dynamic scattering from semiflexible polymers. Macromol., 26 (1993), No 19, 5041-5044. [CrossRef] [Google Scholar]
  7. A. Caspi, M. Elbaum, R. Granek, A. Lachish, D. Zbaida. Semiflexible polymer network: a view from inside. Phys. Rev. Lett., 80 (1998), No 5, 1106-1109. [CrossRef] [Google Scholar]
  8. F. Amblard, A. C. Maggs, B. Yurke, A. N. Pargellis, S. Liebler Subdiffusion and anomalous local viscoelasticity in acting networks. Phys. Rev. Lett., 77 (1996), No 21, 4470-4473. [CrossRef] [PubMed] [Google Scholar]
  9. S. F. Edwards, D. R. Wilkinson. The surface statistics of a granular aggregate. Proc. R. Soc. London A, 381 (1982), No 1780, 17-31. [Google Scholar]
  10. B. H. Zimm. Dynamics of polymer molecules in dilute solution: viscoelasticity flow birefringence and dielectric loss. J. Chem. Phys., 24 (1956), No 2, 269-278. [CrossRef] [Google Scholar]
  11. P. E. Rouse. A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys., 21 (1953), No 7, 1272-1280. [CrossRef] [Google Scholar]
  12. E. Freyssingeas, D. Roux, F. Nallet. Quasi-Elastic Light Scattering Study of Highly Swollen Lamellar and “Sponge" Phases. J. Phys. II France, 7 (1997), 913-929. [CrossRef] [EDP Sciences] [Google Scholar]
  13. E. Helfer, S. Harlepp, L. Bourdieu, J. Robert, F. C. MacKintosh, D. Chatenay. Microrheology of biopolymer-membrane complexes. Phys. Rev. Lett., 85 (2000), No 2, 457-460. [CrossRef] [PubMed] [Google Scholar]
  14. R. Granek, J. Klafter. Anomalous motion of membranes under a localized external potential. Europhys. Lett., 56 (2001), No 1, 15-21. [CrossRef] [Google Scholar]
  15. A. G. Zilman, R. Granek. Membrane dynamics and structure factor. Chem. Phys., 284 (2002), 195-204. [CrossRef] [Google Scholar]
  16. A. G. Zilman, R. Granek. Dynamics of fractal sol-gel polymeric clusters. Phys. Rev. E, 58 (1998), No 3, R2725-R2728. [CrossRef] [Google Scholar]
  17. P. C. Searson, R. Li, K. Sieradzki. Surface Diffusion in the Solid-on-Solid Model. Phys. Rev. Lett. 74 (1995), No 8, 1395-1398. [CrossRef] [PubMed] [Google Scholar]
  18. J. Krug, H. T. Dobb. Anomalous Tracer Diffusion on Surfaces. Phys. Rev. Lett. 76 (1996), No 21, 4096-4096. [CrossRef] [PubMed] [Google Scholar]
  19. S. N. Majumdar, A. Bray. Spatial persistence of fluctuating interfaces. Phys. Rev. Lett. 86 (2001), No 17, 3700-3703. [CrossRef] [PubMed] [Google Scholar]
  20. For a review on fluctuating interfaces, see J. Krug. Origins of scale invariance in growth processes . Adv. Phys., 46 (1997), No 2, 139-282 and [Google Scholar]
  21. J. Krug in Scale Invariance, Interfaces and Non-Equilibrium Dynamics. edited by A. McKane et al. Plenum, New York, 1995. [Google Scholar]
  22. J. Krug, H. Kallabis, S. N. Majumdar, S. J. Cornell, A. J. Bray, C. Sire. Persistence exponents for fluctuating interfaces. Phys. Rev. E, 56 (1997), No 3, 2702-2712. [CrossRef] [Google Scholar]
  23. Z. Toroczkai, E. D. Williams. Nanoscale fluctations at solid surfaces. Phys. Today, 52 (1998), No.12 , 24-29. [Google Scholar]
  24. S. Majaniemi, T. Ala-Nissila, J. Krug. Kinetic roughening of surfaces: Derivation, solution, and application of linear growth equations. Phys. Rev. B, 53 (1995), No 12, 8071-8082. [CrossRef] [Google Scholar]
  25. H. Gao, J. R. Rice. A first order perturbation analysis of crack trapping by arrays of obstacles. J. Appl. Mech., 65 (1989), No 56, 828-836. [CrossRef] [Google Scholar]
  26. J. F. Joanny, P. G. de Gennes. A model for contact angle hysteresis. J. Chem. Phys., 81 (1984), 552-549. [CrossRef] [Google Scholar]
  27. F. Mainardi, P. Pironi. The Fractional Langevin Equation: Brownian Motion Revisited. Extr. Math. 10 (1996), No 1, 140-154; [Google Scholar]
  28. F. Mainardi, A. Mura and F. Tampieri. Brownian motion and anomalous diffusion revisited via a fractional Langevin equation. Modern Problems of Statistical Physics 8 (2009), 3-23. [Google Scholar]
  29. E. Lutz. Fractional Langevin equation. Phys. Rev. E, 64 (2001), No 5, 051106-1-4. [CrossRef] [Google Scholar]
  30. E. Lutz. Fractional Langevin Eqaution. In Fractional Dynamics. Recent Advances. edited by J. Klafter, S. C. Lim and R. Metzler. World Scientific, Singapore, 2012. Ch.12, pp. 285-305. [Google Scholar]
  31. B. B. Mandelbrot, J. W. Van Ness, Fractional Brownian Motions, Fractional Noises and Applications. SIAM Rev., 10 (1968), 422-437. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  32. S. C. Kou, X. S. Xie. Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys. Rev. Lett., 93 (2004), No 18, 180603-1-4. [Google Scholar]
  33. S. C. Kou. Stochastic Modeling in Nanoscale Biophysics: Subdiffusion within Proteins. Annals Applied Statistics, 2 (2008), No 2, 501-535. [CrossRef] [Google Scholar]
  34. S. Burov, E. Barkai. Critical exponent of the fractional Langevin equation. Phys. Rev. Lett., 100 (2008), No 7, 070601-1-4. [CrossRef] [PubMed] [Google Scholar]
  35. K. Sau Fa. Generalized Langevin equation with fractional derivative and long-time correlation function. Phys. Rev. E, 73 (2006), No 6, 061104-1-4. [Google Scholar]
  36. K. Sau Fa. Fractional Langevin equation and Riemann-Liouville fractional derivative. Eur. Phys. J. E, 24 (2007), No 2, 139-143. [CrossRef] [EDP Sciences] [Google Scholar]
  37. I. Goychuk. Viscoelastic Subdiffusion: Generalized Langevin Equation Approach. Adv. Chem. Phys., 150 (2012), 187-253. [CrossRef] [Google Scholar]
  38. A. Taloni, M. A. Lomholt. Langevin formulation for single-file diffusion. Phys. Rev. E, 78 (2008), No 5, 051116-1-8. [CrossRef] [Google Scholar]
  39. L. Lizana, T. Ambjornsson, A. Taloni, E. Barkai, M. A. Lomholt. et al.. Foundation of fractional Langevin equation: Harmonization of a many-body problem. Phys. Rev. E, 81 (2010), No 5, 051118-1-8. [CrossRef] [Google Scholar]
  40. D. Panja. Generalized Langevin equation formulation for anomalous polymer dynamics. J. Stat. Mech., (2010), L02001-1-8. D. Panja. Anomalous polymer dynamics is non-Markovian: memory effects and the generalized Langevin equation formulation. J. Stat. Mech., (2010), P06011-1-34. [Google Scholar]
  41. A. Taloni, A. Chechkin, J. Klafter. Correlations in a generalized elastic model: Fractional Langevin equation approach. Phys. Rev. E, 82 (2010), No 6, 061104-1-15. [CrossRef] [Google Scholar]
  42. C. Fox. The G and H Functions as symmetrical Fourier kernels. Trans. Amer. Math. Soc., 98 (1961), 395-429. [MathSciNet] [Google Scholar]
  43. A. M. Mathai, R. K. Saxena. The H-Function with Application in Statistics and Other Discplines. Wiley Eastern Limited, New Delhi-Bangalore-Bombay, 1978. [Google Scholar]
  44. R. Hilfer (Ed.). Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. [Google Scholar]
  45. W. G. Gloeckle, T. F. Nonnenmacher. Fractional Integral Operators and Fox Functions in the Theory of Viscoelasticity. Macromolecules, 24 (1991), 6426-6434. [CrossRef] [Google Scholar]
  46. G. Gloeckle, T. F. Nonnenmacher. Fox Function Representation of Non-Debye Relaxation Processes. Journ. Stat. Phys. 71 (1993), Nos 3/4, 741-757. [Google Scholar]
  47. R. Metzler, W. G. Gloeckle, T. F. Nonnenmacher, Fractional model equation for anomalous diffusion. Physica A, 211 (1994), 13-24. [Google Scholar]
  48. R. Metzler, J. Klafter. The Random Walk Guide to Anomalous Difffusion: a Fractional Dynamics Approach. Phys. Rep. 339 (2000), 1-77. [NASA ADS] [CrossRef] [Google Scholar]
  49. F. Mainardi, G. Pagnini, R. K. Saxena. Fox H functions in fractional diffusion. Journ. Comput. Applied Math., 178 (2005), Nos 1-2, 321-331. [CrossRef] [Google Scholar]
  50. S. I. Denisov, S. B. Yuste, Yu. S. Bystrik, H. Kantz, K. Lindenberg. Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions. Phys. Rev. E, 84 (2011), 061143-1-7. [CrossRef] [Google Scholar]
  51. R. K. Saxena, A. M. Mathai, H. J. Haubold. Fractional Reaction-Diffusion Equations. Astrophys Space Sci, 305 (2006), 289-296. [Google Scholar]
  52. M. O. Vlad, R. Metzler, J. Ross. Generalized Huber kinetics for nonlinear rate processes in disordered systems: Nonlinear analogs of stretched exponential. Phys. Rev. E, 57 (1998), No. 6, 6497-6505. [CrossRef] [Google Scholar]
  53. N. Laskin. Fractional quantum mechanics and Levy path integrals. Phys. Lett. A, 268 (2000), 298-305. [CrossRef] [MathSciNet] [Google Scholar]
  54. N. Laskin. Fractals and quantum mechanics. Chaos, 10 (2000), No. 4, 780-790. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  55. N. Laskin. Fracional quantum mechanics. Phys. Rev. E 62, (2000), No.3, 3135-3145. N. Laskin. Levy flights over quantum paths. Communications Nonlinear Sci and Numer. Simulations, 12 (2007), No. 1, 2-18. [Google Scholar]
  56. J. Dong. Green’s function for the time-dependent scattering problem in the fractional quantum mechanics. Journ. Math. Phys., 52 (2011), 042103-1-10. [CrossRef] [Google Scholar]
  57. A. A. Kilbas, M. Saigo, H-Transforms. Theory and Applications. Chapman and Hall, London, 2004. [Google Scholar]
  58. A. M. Mathai, R. K. Saxena, H. J. Haubold, The H - Function. Theory and Applications. Springer, New York, 2010. [Google Scholar]
  59. A. P. Prudnikov, Y. A. Brychkov, O. I. Marichev. Integral and Series. Vol.3: More special Functions. Gordon and Breach Science, Amsterdam, 1990. [Google Scholar]
  60. A. Taloni, A. Chechkin, J. Klafter. Unusual response to a localized perturbation in a generalized elastic model. Phys. Rev. E, 84 (2011), No 2, 021101-1-7. [CrossRef] [Google Scholar]
  61. E. Helfer, S. Harlepp, L. Bourdieu, J. Robert, F. C. MacKintosh, D. Chatenay. Microrheology of biopolymer-membrane complexes. Phys. Rev. Lett., 85 (2000), No 2, 457-460. [CrossRef] [PubMed] [Google Scholar]
  62. E. Helfer, S. Harlepp, L. Bourdieu, J. Robert, F. C. MacKintosh, D. Chatenay. Viscoelastic properties of actin-coated membranes. Phys. Rev. E, 63 (2001), No 2, 021904-1-13. [CrossRef] [Google Scholar]
  63. Chau-Hwang Leef et al. Three-Dimensional Characterization of Active Membrane Waves on Living Cells. Phys. Rev. Lett. 103 (2009), No 23, 238101-1-4. [Google Scholar]
  64. I. Podlubny. Fractional Differential Equations. Academic Press, New York, 1999. [Google Scholar]
  65. M. Caputo. Linear model of dissipation whose Q is almost frequency independent. Geophys. J. R. Astr. Soc., 13 (1967), 529-539. [Google Scholar]
  66. D. C. Champeney. Fourier Transforms and Physical Applications. Academic Press, London, 1973. [Google Scholar]
  67. A. Taloni, A. Chechkin, J. Klafter. Generalized elastic model: Thermal vs. non-thermal initial conditions. Universal scaling, roughening, ageing and ergodicity. Europhys. Lett., 97 (2012), No 3, 30001-p1-p6. [CrossRef] [Google Scholar]
  68. A. L. Barabasi, H. E. Stanley. Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge, 1994. [Google Scholar]
  69. P. Meakin. Fractal and scaling growth far from equilibrium. Cambridge University Press, Cambridge, 1998. [Google Scholar]
  70. F. Family, T. Vicsek. Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model. J. Phys. A: Math. Gen., 18 (1985), No 2, L75-L81. [CrossRef] [Google Scholar]
  71. M. Abramowitz, I. Stegun. Handbook of Mathematical Functions. Dover, New York, 1964. [Google Scholar]
  72. A. P. Prudnikov, Y. A. Brychkov, O. I. Marichev. Integrals and Series. Vol.I: Elementary Functions. New York: Gordon and Breach, 1986. [Google Scholar]
  73. I. M. Gelfand, G. E. Shilov, Generalized Functions. Academic Press, 1964. [Google Scholar]
  74. A. W. J. Erdelyi. Asymptotic Expansions. Dover, New York, 1956. [Google Scholar]
  75. R. Kubo, M. Toda, N. Hatsushime. Statistical Physics II: Nonequilibrium Statistical Mechanics. Springer, Berlin, 1991. [Google Scholar]
  76. U. M. B. Marconi, A. Puglisi, L. Rondoni, A. Vulpiani. Fluctuation-dissipation: response theory in statistical physics. Phys. Rep., 461 (2008), No 4-6, 111-195. [CrossRef] [Google Scholar]
  77. D. Villamaina, A. Baldassarri, A. Puglisi, A. Vulpiani. The fluctuation-dissipation relation: how does one compare correlation functions and responses?. J. Stat. Mech., (2009), P07024-1-22. [Google Scholar]
  78. E. Barkai, R. Silbey. Theory of Single File Diffusion in a Force Field. Phys. Rev. Lett., 102 (2009), No 5, 050602-1-4. [CrossRef] [PubMed] [Google Scholar]
  79. I. S. Gradshtein, I. M. Rizhikl. Tables of Integrals. Series and Products. Academic Press, New York, 2007. [Google Scholar]
  80. R. Santachiara, A Rosso, W Krauth. Universal width distributions in non-Markovian Gaussian processes. J. Stat. Mech., (2007), P02009–. [Google Scholar]
  81. G. H. Hardy. Divergent Series. Clarendon Press, Oxford, 1949. [Google Scholar]
  82. F. W. J. Olver. Asymptotics and Special Functions. Academic Press, New York, 1974. [Google Scholar]

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