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Anomalous Diffusion Equations with Multiplicative Acceleration

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Abstract

A generalization of the model of Lévy walks with traps is considered. The main difference between the model under consideration and the already existing models is the introduction of multiplicative particle acceleration at collisions. The introduction of acceleration transfers the consideration of walks to coordinate–momentum phase space, which allows both the spatial distribution of particles and their spectrum to be obtained. The kinetic equations in coordinate–momentum phase space have been derived for the case of walks with two possible states. This system of equations in a special case is shown to be reduced to ordinary Lévy walks. This system of kinetic equations admits of integration over the spatial variable, which transfers the consideration only to momentum space and allows the spectrum to be calculated. An exact solution of the kinetic equations can be obtained in terms of the Laplace–Mellin transform. The inverse transform can be performed only for the asymptotic solutions. The calculated spectra are compared with the results of Monte Carlo simulations, which confirm the validity of the derived asymptotics.

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References

  1. E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965).

    Article  ADS  Google Scholar 

  2. H. Scher and M. Lax, Phys. Rev. B 7, 4491 (1973).

    Article  ADS  MathSciNet  Google Scholar 

  3. H. Scher and M. Lax, Phys. Rev. B 7, 4502 (1973).

    Article  ADS  Google Scholar 

  4. J. Klafter, A. Blumen, and M. F. Shlesinger, Phys. Rev. A 35, 3081 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  5. V. M. Zolotarev, V. V. Uchaikin, and V. V. Saenko, J. Exp. Theor. Phys. 88, 780 (1999).

    Article  ADS  Google Scholar 

  6. V. Yu. Zaburdaev and K. V. Chukbar, J. Exp. Theor. Phys. 94, 252 (2002).

    Article  ADS  Google Scholar 

  7. V. V. Uchaikin and I. V. Yarovikova, Comput. Math. Math. Phys. 43, 1478 (2003).

    MathSciNet  Google Scholar 

  8. V. V. Uchaikin, JETP Lett. 91, 105 (2010).

    Article  ADS  Google Scholar 

  9. M. F. Shlesinger, B. J. West, and J. Klafter, Phys. Rev. Lett. 58, 1100 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  10. V. V. Uchaikin, Phys. A (Amsterdam, Neth.) 255, 65 (1998).

    Article  ADS  Google Scholar 

  11. D. Froemberg, M. Schmiedeberg, E. Barkai, and V. Yu. Zaburdaev, Phys. Rev. E 91, 022131 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  12. V. V. Saenko, Phys. A (Amsterdam, Neth.) 444, 765 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  13. V. Yu. Zaburdaev, J. Stat. Phys. 123, 871 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  14. I. M. Sokolov and R. Metzler, Phys. Rev. E 67, 010101(R) (2003).

    Article  ADS  Google Scholar 

  15. V. V. Uchaikin and R. T. Sibatov, Grav. Cosmol. 18, 122 (2012).

    Article  ADS  Google Scholar 

  16. V. V. Uchaikin and R. T. Sibatov, J. Phys. A 44, 145501 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  17. V. V. Uchaikin, Phys. Usp. 56, 1074 (2013).

    Article  ADS  Google Scholar 

  18. V. V. Uchaikin, R. T. Sibatov, and A. N. Byzykchi, Bull. Russ. Acad. Sci.: Phys. 79, 592 (2015).

    Article  Google Scholar 

  19. V. V. Uchaikin, R. T. Sibatov, and A. N. Byzykchi, Comm. Appl. Ind. Math. 6, e–479 (2014).

    Google Scholar 

  20. V. V. Uchaikin and R. T. Sibatov, Tech. Phys. Lett. 30, 316 (2004).

    Article  ADS  Google Scholar 

  21. M. Bologna, G. Ascolani, and P. Grigolini, J. Math. Phys. 51, 043303 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  22. V. V. Uchaikin and R. T. Sibatov, J. Exp. Theor. Phys. 109, 537 (2009).

    Article  ADS  Google Scholar 

  23. D. Froemberg and E. Barkai, Eur. Phys. J. B 86, 331 (2013).

    Article  ADS  Google Scholar 

  24. C. Godrèche and J. M. Luck, J. Stat. Phys. 104, 489 (2001).

    Article  Google Scholar 

  25. M. Magdziarz, W. Szczotka, and P. Zebrowski, J. Stat. Phys. 147, 74 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  26. M. Magdziarz, H. P. Scheffler, P. Straka, and P. Zebrowski, Stoch. Processes Appl. 125, 4021 (2015).

    Article  Google Scholar 

  27. M. Magdziarz and T. Zorawik, Phys. Rev. E 94, 022130 (2016).

    Article  ADS  Google Scholar 

  28. M. Magdziarz and T. Zorawik, Commun. Nonlin. Sci. Numer. Simul. 48, 462 (2017).

    Article  Google Scholar 

  29. M. Magdziarz, Phys. Rev. E 95, 022126 (2017).

    Article  ADS  Google Scholar 

  30. A. Milovanov and L. Zelenyi, Phys. Rev. E 64, 052101 (2001).

    Article  ADS  Google Scholar 

  31. V. Yu. Zaburdaev, M. Schmiedeberg, and H. Stark, Phys. Rev. E 78, 1 (2008).

    Article  Google Scholar 

  32. V. Zaburdaev, S. I. Denisov, and P. Hanggi, Phys. Rev. Lett. 106, 16 (2011).

    Google Scholar 

  33. S. I. Denisov, V. Zaburdaev, and P. Hanggi, Phys. Rev. E 85, 1 (2012).

    Google Scholar 

  34. V. Yu. Zaburdaev, S. I. Denisov, and J. Klafter, Rev. Mod. Phys. 87, 483 (2015).

    Article  ADS  Google Scholar 

  35. S. A. Trigger, W. Ebeling, G. J. F. van Heijst, et al., Phys. Plasmas 17, 042102 (2010).

    Article  ADS  Google Scholar 

  36. A. A. Dubinova and S. A. Trigger, Phys. Lett. A 376, 1930 (2012).

    Article  ADS  Google Scholar 

  37. V. L. Ginzburg, Usp. Fiz. Nauk 62 (2), 37 (1957).

    Article  Google Scholar 

  38. V. L. Ginzburg and S. I. Syrovatskii, Sov. Phys. Usp. 3, 504 (1960).

    Article  Google Scholar 

  39. V. L. Ginzburg and S. I. Syrovatskii, Origin of Cosmic Rays (Akad. Nauk SSSR, Moscow, 1963; Pergamon, Oxford, 1964).

    Book  Google Scholar 

  40. P. Duffy, J. G. Kirk, Y. A. Gallant, and R. O. Dendy, Astron. Astrophys. 302, L21 (1995).

    ADS  Google Scholar 

  41. B. R. Ragot and J. G. Kirk, Astron. Astrophys. 327, 432 (1997).

    ADS  Google Scholar 

  42. G. Zimbardo, P. Pommois, and P. Veltri, Astrophys. J. 639, L91 (2006).

    Article  ADS  Google Scholar 

  43. S. Perri and G. Zimbardo, Astrophys. J. 671, L177 (2007).

    Article  ADS  Google Scholar 

  44. S. Perri and G. Zimbardo, J. Geophys. Res. 113, A03107 (2008).

    Article  ADS  Google Scholar 

  45. S. Perri and G. Zimbardo, Astrophys. J. 693, L118 (2009).

    Article  ADS  Google Scholar 

  46. S. Perri and G. Zimbardo, Adv. Space Res. 44, 465 (2009).

    Article  ADS  Google Scholar 

  47. T. Sugiyama and D. Shiota, Astrophys. J. 731, L34 (2011).

    Article  ADS  Google Scholar 

  48. E. M. Trotta and G. Zimbardo, Astron. Astrophys. 530, A130 (2011).

    Article  ADS  Google Scholar 

  49. S. Perri, E. Amato, and G. Zimbardo, Astron. Astrophys. 596, A34 (2016).

    Article  ADS  Google Scholar 

  50. B. R. Ragot, Astrophys. J. 525, 524 (1999).

    Article  ADS  Google Scholar 

  51. B. R. Ragot, Astrophys. J. 647, 630 (2006).

    Article  ADS  Google Scholar 

  52. B. R. Ragot, Astrophys. J. 645, 1169 (2006).

    Article  ADS  Google Scholar 

  53. G. Zimbardo, S. Perri, P. Pommois, and P. Veltri, Adv. Space Res. 49, 1633 (2012).

    Article  ADS  Google Scholar 

  54. N. Sánchez and E. J. Alfaro, Astrophys. J. Suppl. Ser. 178, 1 (2008).

    Article  ADS  Google Scholar 

  55. N. Schneider, S. Bontemps, P. Girichidis, et al., Mon. Not. R. Astron. Soc. Lett. 453, L41 (2015).

    Article  ADS  Google Scholar 

  56. B. F. Elmegreen, Publ. Astron. Soc. Austral. 15, 74 (2013).

    Article  ADS  Google Scholar 

  57. R. de la Fuente Marcos and C. de la Fuente Marcos, Astron. Astrophys. 452, 163 (2006).

    Article  ADS  Google Scholar 

  58. A. A. Lagutin, Probl. At. Sci. Technol. 6, 214 (2001).

    Google Scholar 

  59. A. D. Erlykin, A. A. Lagutin, and A. W. Wolfendale, Astropart. Phys. 19, 351 (2003).

    Article  ADS  Google Scholar 

  60. H. A. Kermani and J. Fatemi, South Afr. J. Sci. 107, 2 (2011).

    Google Scholar 

  61. N. Ketabi and J. Fatemi, Trans. B: Mech. Eng. 16, 269 (2009).

    Google Scholar 

  62. Yu. E. Litvinenko and F. Effenberger, Astrophys. J. 796, 125 (2014).

    Article  ADS  Google Scholar 

  63. A. A. Lagutin, Yu. A. Nikulin, and V. V. Uchaikin, Nucl. Phys. B (Proc. Suppl.) 97, 267 (2001).

    Article  ADS  Google Scholar 

  64. A. A. Lagutin and A. G. Tyumentsev, Izv. Akad. Nauk, Ser. Fiz. 67, 439 (2003).

    Google Scholar 

  65. A. A. Lagutin and V. V. Uchaikin, Nucl. Instrum. Methods Phys. Res., Sect. B 201, 212 (2003).

    Article  ADS  Google Scholar 

  66. A. A. Lagutin, A. G. Tyumentsev, and A. V. Yushkov, Nucl. Phys. B (Proc. Suppl.) 175–176, 555 (2008).

    Article  Google Scholar 

  67. A. A. Lagutin, A. G. Tyumentsev, and A. V. Yushkov, Bull. Russ. Acad. Sci.: Phys. 71, 590 (2007).

    Article  Google Scholar 

  68. S. Perri and G. Zimbardo, Astrophys. J. 750, 87 (2012).

    Article  ADS  Google Scholar 

  69. A. Lazarian and H. Yan, Astrophys. J. 784, 38 (2014).

    Article  ADS  Google Scholar 

  70. V. V. Uchaikin, J. Exp. Theor. Phys. 116, 897 (2013).

    Article  ADS  Google Scholar 

  71. V. V. Uchaikin, R. T. Sibatov, and V. V. Saenko, J. Phys.: Conf. Ser. 409, 012057 (2013).

    Google Scholar 

  72. V. V. Uchaikin, R. T. Sibatov, and V. V. Saenko, Bull. Russ. Acad. Sci.: Phys. 77, 619 (2013).

    Article  Google Scholar 

  73. G. Zimbardo and S. Perri, Astrophys. J. 778, 35 (2013).

    Article  ADS  Google Scholar 

  74. S. Perri and G. Zimbardo, Astrophys. J. 815, 75 (2015).

    Article  ADS  Google Scholar 

  75. S. Perri, G. Zimbardo, F. Effenberger, and H. Fichtner, Astron. Astrophys. 578, A2 (2015).

    Article  ADS  Google Scholar 

  76. V. V. Uchaikin, R. T. Sibatov, and A. N. Byzykchi, Bull. Russ. Acad. Sci.: Phys. 79, 592 (2015).

    Article  Google Scholar 

  77. V. V. Uchaikin and R. T. Sibatov, J. Phys.: Conf. Ser. 798, 012029 (2017).

    Google Scholar 

  78. V. V. Uchaikin and R. T. Sibatov, Chaos, Solitons Fractals 102, 197 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  79. A. R. Bell, Mon. Not. R. Astron. Soc. 182, 147 (1978).

    Article  ADS  Google Scholar 

  80. E. G. Berezhko and G. F. Krymskii, Sov. Phys. Usp. 31, 27 (1988).

    Article  ADS  Google Scholar 

  81. V. V. Uchaikin, JETP Lett. 92, 200 (2010).

    Article  ADS  Google Scholar 

  82. A. Wandel, D. Eichler, and J. R. Letaw, Astrophys. J. 316, 676 (1987).

    Article  ADS  Google Scholar 

  83. V. M. Zolotarev, One-Dimensional Stable Distributions, Vol. 65 of Translations of Mathematical Monographs (Nauka, Moscow, 1983; Am. Math. Soc., Providence, 1986).

    Book  MATH  Google Scholar 

  84. V. V. Uchaikin, J. Exp. Theor. Phys. 88, 1155 (1999).

    Article  ADS  Google Scholar 

  85. V. S. Berezinskii, S. V. Bulanov, V. L. Ginzburg, et al., Astrophysics of Cosmic Rays, Ed. by V. L. Ginzburg (Nauka, Moscow, 1990; North-Holland, Amsterdam, 1990).

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  1. Kapitza Research Technological Institute, Ulyanovsk State University, ul. L. Tolstogo 42, Ulyanovsk, 432000, Russia

    V. V. Saenko

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Original Russian Text © V.V. Saenko, 2018, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2018, Vol. 153, No. 4, pp. 557–575.

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Saenko, V.V. Anomalous Diffusion Equations with Multiplicative Acceleration. J. Exp. Theor. Phys. 126, 462–478 (2018). https://doi.org/10.1134/S1063776118030202

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