Geodesic
Fast Fermi Acceleration and Entropy Growth
T. Pereira1, 2 ; D. Turaev1, 3
1 Department of Mathematics, Imperial College London, SW7 2AZ, United Kingdom
2 Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, Brazil,
3 Lobachevsky University of Nizhny Novgorod, 603950 Russia
Mathematical modelling of natural phenomena, Tome 10 (2015) no. 3, pp. 31-47 Cet article a éte moissonné depuis la source EDP Sciences

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Fermi acceleration is the process of energy transfer from massive objects in slow motion to light objects that move fast. The model for such process is a time-dependent Hamiltonian system. As the parameters of the system change with time, the energy is no longer conserved, which makes the acceleration possible. One of the main problems is how to generate a sustained and robust energy growth. We show that the non-ergodicity of any chaotic Hamiltonian system must universally lead to the exponential growth of energy at a slow periodic variation of parameters. We build a model for this process in terms of a Geometric Brownian Motion with a positive drift and relate it to the entropy increase.
DOI : 10.1051/mmnp/201510304

T. Pereira 1, 2 ; D. Turaev 1, 3

1 Department of Mathematics, Imperial College London, SW7 2AZ, United Kingdom
2 Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, Brazil,
3 Lobachevsky University of Nizhny Novgorod, 603950 Russia 
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T. Pereira; D. Turaev. Fast Fermi Acceleration and Entropy Growth. Mathematical modelling of natural phenomena, Tome 10 (2015) no. 3, pp. 31-47. doi: 10.1051/mmnp/201510304

[1] D.V. Anosov Izv. Akad. Nauk SSSR, Ser. Mat. 1960 721

[2] B. Batistić Phys. Rev. E 2014 032909

[3] B. Batistić, M. Robnik J. Phys. A: Math. Theor. 2011 365101

[4] J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A.J. Sierk, W.J. Swiatecki Ann. Phys. (N.Y.) 1978 330

[5] M.V.S. Bonanca, M.A.M. De Aguiar Physica A 2006 333

[6] R. Brown, E. Ott, C. Grebogi J. Stat. Phys. 1987 511

[7] A. Canergie, I.C. Percival J. Phys. A 1984 801

[8] R.E. De Carvalho, F.C. De Souza, E.D. Leonel J. Phys. A: Math. Theor. 2006 3561

[9] R.E. De Carvalho, F.C. De Souza, E.D. Leonel Phys. Rev. E 2006 066229

[10] E.V. Derishev, V.V. Kocharovsky, Vl.V. Kocharovsky Physics - Uspekhi 2007 308

[11] A. Dovbysh J. Appl. Math. Mech. 1992 188

[12] E. Fermi Phys. Rev. 1949 1169

[13] E. Forest, R.D. Ruth Physica D 1990 105

[14] V. Gelfreich, V. Rom-Kedar, K. Shah, D. Turaev Phys. Rev. Lett. 2011 074101

[15] V. Gelfreich, V. Rom-Kedar, D. Turaev Chaos 2012 033116

[16] V. Gelfreich, V. Rom-Kedar, D. Turaev J. Phys. A 2014 395101

[17] S. Hilbert, P. Hänggi, J. Dunkel Phys. Rev. E 2014 062116

[18] C. Jarzynski Phys. Rev. E 1993 4340

[19] S. Laederich, M. Levi Invariant curves and time-dependent potentials, Ergod Th. & Dynam. Sys. 1991 365

[20] F. Lenz, F.K. Diakonos, P. Schmelcher Phys. Rev. Lett. 2008 014103

[21] T. Kasuga Proc. Jpn. Acad. 1961 366

[22] E.D. Leonel, L.A. Bunimovich Phys. Rev. Lett. 2010 224101

[23] E.D. Leonel, D.F.M. Oliveira, A. Loskutov Chaos 2009 033142

[24] M.A. Lieberman, V.A. Godyak IEEE Trans. Plasma Sci. 1998 955

[25] P. Lochak, C. Meunier. Multiphase Averaging for Classical Systems. Springer-Verlag, New York, 1988.

[26] A. Loskutov, A. Ryabov J. Stat. Phys. 2002 995

[27] A. Loskutov, A.B. Ryabov, L.G. Akinshin JETP 1999 966

[28] A. Loskutov, A.B. Ryabov, L.G. Akinshin J. Phys. A: Math. Gen. 2000 7973

[29] R.S. MacKay. Nonlinear Dynamics and Chaos: Advances and Perspectives. Springer, 2010, pp. 89–102.

[30] B.K. Oksendal, Stochastic Differential Equations: An Introduction with Applications. Springer, 2002.

[31] D.F.M. Oliveira, M. Robnik Phys. Rev. E 2011 026202

[32] D.F.M. Oliveira, J. Vollmer, E.D. Leonel Physica D 2011 389

[33] E. Ott Phys. Rev. Lett. 1979 1628

[34] T. Pereira, D. Turaev Phys. Rev. E 2015 010901(R)

[35] O. Peters, W. Klein Phys. Rev. Lett. 2013 100603

[36] L.D. Pustyl’Nikov Theor. Math. Phys. 1983 1035

[37] L.D. Pustyl’Nikov Sb. Math. 1995 231

[38] K. Shah Phys. Rev. E 2011 046215

[39] K. Shah, D. Turaev, V. Rom-Kedar Phys. Rev. E 2010 056205

[40] D. Turaev, Exponential Fermi acceleration in adiabatically perturbed Hamiltonian systems. In: Proceedings of the 8th European Nonlinear Dynamics Conference (ENOC 2014). 2014.

[41] D. Turaev, V. Rom-Kedar Nonlinearity 1998 575

[42] S.M. Ulam, On some statistical properties of dynamical systems. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3. University of California Press, Berkeley, 1961, 315–320.

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