theory and studied the criteria for chaos by stating with super-Hamiltonian of an
electron. In this case, the resonance condition gives the threshold of the Chirikov
resonance-overlap criterion for the onset of chaotic motion in deterministic Hamiltonian systems. The theory was extended by Rax to relating to the relativistic laser
electron interaction in the configuration of the counter-propagating two laser problem [29]. In the papers, Hamilton-Jacobi equation is derived to solve the perturbation
equation. However, it is difficult to explain these methods without complicated
mathematics.
References
1. J. Meyer-ter-Vehn, Z.M. Sheng, Phys. Plasmas 6, 641 (1999)
2. S.C. Wilks, W.L. Kruer, IEEE J. Quant. Elec 33, 1953 (1997)
3. T. Nakamura et al., Phys. Plasmas 9, 1801 (2002)
4. M. Tanimoto et al., Phys. Rev. E 68, 026401 (2003)
5. Z.M. Sheng et al., Phys. Rev. Lett. 88, 055004 (2002)
6. S. Rassou, A. Bourdier, M. Drouin, Phys. Plasma 21, 083101 (2014)
7. C. Tsallis, J. Stat. Phys. 52, 479 (1988)
8. J. Klafter, M. F. Sclesinger, and G. Zumofen, Physics Today, Feb. p.33 (1996)
9. B. Mandelbrot, The Fractal Geometry of Nature, (Freeman, 1982)
10. A.V. Milovanov, J.J. Rasmussen, Phys. Lett. A 378, 1492 (2014)
11. S.V. Bulanov et al., Aust. J. Plant Physiol. 83, 905830202 (2017)
12. E. Barkai, R. Metzler, J. Klafter, Phys. Rev. E 61, 132 (2000)
13. J. Anderson et al., Phys. Plasmas 21, 122109 (2014)
14. C. Tsallis et al., Phys. Rev. Lett. 75, 3589 (1995)
15. A. Bourdier, M. Drouin, Laser Part. Beams 27, 545 (2009)
16. Y. Kuramitsu et al., Phys. Plasmas 18, 010701 (2011)
17. S. Kojima et al., Commun. Phys. 2, 99 (2019)
18. E. Fermi, Phys. Rev. 75, 1169 (1949); Astrophys. J. 119, 1. (1954)
19. A. R. Bell, Mon. Not. R. Astro. Soc. 182, 147 (1978); K. M. Schure et al., Space Sci. Rev.
173, 491 (2012)
20. P. Chen et al., Phys. Rev. Lett. 89, 161101 (2002)
21. M. Hoshino, Astrophys. J. 672, 940 (2008)
22. M. Iwamoto et al., Astrophys. J. 840, 52 (2017)
23. L. Sironi, A.Y. Spitkovsy, J. Arons, Astrophys. J. 771, 54 (2013)
24. S. Moradi et al., Physical Rev. Research 2, 013027 (2020)
25. S. Bulanov et al., Phys. Plasmas 22, 063108 (2015)
26. U. Tilnakli, E.P. Borges, Sci. Rep. 6, 23644 (2016)
27. A.J. Lichtenberg, M.A. Liberman, Regular and Stochastic Motion (Springer, 1983). Chapter 3
28. T. Mendoca, Phys. Rev. A, 3592 (1983)
29. J.M. Rax, Phys. Fluids B-4, 3962 (1992)
References
371
electron. In this case, the resonance condition gives the threshold of the Chirikov
resonance-overlap criterion for the onset of chaotic motion in deterministic Hamiltonian systems. The theory was extended by Rax to relating to the relativistic laser
electron interaction in the configuration of the counter-propagating two laser problem [29]. In the papers, Hamilton-Jacobi equation is derived to solve the perturbation
equation. However, it is difficult to explain these methods without complicated
mathematics.
References
1. J. Meyer-ter-Vehn, Z.M. Sheng, Phys. Plasmas 6, 641 (1999)
2. S.C. Wilks, W.L. Kruer, IEEE J. Quant. Elec 33, 1953 (1997)
3. T. Nakamura et al., Phys. Plasmas 9, 1801 (2002)
4. M. Tanimoto et al., Phys. Rev. E 68, 026401 (2003)
5. Z.M. Sheng et al., Phys. Rev. Lett. 88, 055004 (2002)
6. S. Rassou, A. Bourdier, M. Drouin, Phys. Plasma 21, 083101 (2014)
7. C. Tsallis, J. Stat. Phys. 52, 479 (1988)
8. J. Klafter, M. F. Sclesinger, and G. Zumofen, Physics Today, Feb. p.33 (1996)
9. B. Mandelbrot, The Fractal Geometry of Nature, (Freeman, 1982)
10. A.V. Milovanov, J.J. Rasmussen, Phys. Lett. A 378, 1492 (2014)
11. S.V. Bulanov et al., Aust. J. Plant Physiol. 83, 905830202 (2017)
12. E. Barkai, R. Metzler, J. Klafter, Phys. Rev. E 61, 132 (2000)
13. J. Anderson et al., Phys. Plasmas 21, 122109 (2014)
14. C. Tsallis et al., Phys. Rev. Lett. 75, 3589 (1995)
15. A. Bourdier, M. Drouin, Laser Part. Beams 27, 545 (2009)
16. Y. Kuramitsu et al., Phys. Plasmas 18, 010701 (2011)
17. S. Kojima et al., Commun. Phys. 2, 99 (2019)
18. E. Fermi, Phys. Rev. 75, 1169 (1949); Astrophys. J. 119, 1. (1954)
19. A. R. Bell, Mon. Not. R. Astro. Soc. 182, 147 (1978); K. M. Schure et al., Space Sci. Rev.
173, 491 (2012)
20. P. Chen et al., Phys. Rev. Lett. 89, 161101 (2002)
21. M. Hoshino, Astrophys. J. 672, 940 (2008)
22. M. Iwamoto et al., Astrophys. J. 840, 52 (2017)
23. L. Sironi, A.Y. Spitkovsy, J. Arons, Astrophys. J. 771, 54 (2013)
24. S. Moradi et al., Physical Rev. Research 2, 013027 (2020)
25. S. Bulanov et al., Phys. Plasmas 22, 063108 (2015)
26. U. Tilnakli, E.P. Borges, Sci. Rep. 6, 23644 (2016)
27. A.J. Lichtenberg, M.A. Liberman, Regular and Stochastic Motion (Springer, 1983). Chapter 3
28. T. Mendoca, Phys. Rev. A, 3592 (1983)
29. J.M. Rax, Phys. Fluids B-4, 3962 (1992)
References
371