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velocity law of the moving wall such curves exist, while they do not for sawtooth velocity law that is discontinuous. Consequently, at the first case particles cannot accelerate infinitely, reversely to what happens at the last one.
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exist. These invariant curves act as barriers that do not allow for a particle to further accelerate and the average velocity of a population of particles saturates after finite iterations of the map. For instance, for
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Barnett A., Cohen D., Heller E.J. (2000). "Deformations and
Dilations of Chaotic Billiards: Dissipation Rate, and Quasiorthogonality of the Boundary Wave Functions".
551:
Strongly chaotic billiard with oscillating boundary can serve as a paradigm for driven chaotic systems. In the experimental arena this topic arises in the theory of
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A.P. Itin, A.I. Neishtadt (2012), Fermi acceleration in time-dependent rectangular billiards due to multiple passages through resonances, Chaos 22, 026119.
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L. D. Pustyl'nikov (1995). "Poincaré models, rigorous justification of the second law of thermodynamics from mechanics, and Fermi acceleration mechanism".
459:
223:
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Cohen D (2000). "Chaos and Energy
Spreading for Time-Dependent Hamiltonians, and the Various Regimes in the Theory of Quantum Dissipation".
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Though the 1D Fermi–Ulam model does not lead to acceleration for smooth oscillations, unbounded energy growth has been observed in 2D
858:, in Proceedings of the 38th Karpacz Winter School of Theoretical Physics, Edited by P. Garbaczewski and R. Olkiewicz (Springer, 2002).
727:
A.P. Itin, A.I. Neishtadt, A.A Vasiliev (2001), Resonant phenomena in slowly perturbed rectangular billiards, Phys. Lett. A 291, 133.
121:
959:
Friedman N., Kaplan A., Carasso D., Davidson N. (2001). "Observation of
Chaotic and Regular Dynamics in Atom-Optics Billiards".
628:
L. D. Pustyl'nikov (1988). "A new mechanism for particle acceleration and a relativistic analogue of the Fermi-Ulam model".
559:. The driving induces diffusion in energy, and consequently the absorption coefficient is determined by the Kubo formula.
671:
Loskutov A., Ryabov A. B., Akinshin L. G. (2000). "Properties of some chaotic billiards with time-dependent boundaries".
525:
575:
L.D. Pustyl'nikov, (1983). On a problem of Ulam. Teoret. Mat.Fiz.57, 128-132. Engl. transl. in Theor. Math. Phys. 57.
1349:
528:, then under some general conditions the energy of the particle tends to infinity for an open set of initial data.
783:
F. Lenz; F. K. Diakonos; P. Schmelcher (2008). "Tunable Fermi
Acceleration in the Driven Elliptical Billiard".
1332:: A widely acknowledged scientific book that treats FUM, written by A. J. Lichtenberg and M. A. Lieberman (
521:
of the particle are bounded) was given first by L. D. Pustyl'nikov in (see also and references therein).
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71:
A. J. Lichtenberg and M. A. Lieberman provided a simplified version of FUM (SFUM) that derives from the
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Barnett A., Cohen D., Heller E.J. (2001). "Rate of energy absorption for a driven chaotic cavity".
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In spite of these negative results, if one considers the Fermi–Ulam model in the framework of the
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R. Brown; E. Ott; C. Grebogi (1987). "Ergodic
Adiabatic Invariants of Chaotic systems".
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972:
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Blocki J., Boneh Y., Nix J.R., Randrup J., Robel M., Sierk A.J., Swiatecki W.J. (1978).
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between a fixed wall and a moving one, each of infinite mass. The walls represent the
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Wilkinson M (1988). "Statistical aspects of dissipation by Landau-Zener transitions".
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Gelfreich V., Turaev D. (2008). "Fermi acceleration in non-autonomous billiards".
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If the velocity law of the moving wall is differentiable enough, according to
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309:{\displaystyle \varphi _{n+1}=\varphi _{n}+{\frac {kM}{u_{n+1}}}{\pmod {k}},}
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555:, and more recently in the studies of cold atoms that are trapped in
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billiards is found to be much larger than that in billiards that are
518:
28:
844:
797:
517:
The rigorous solution of the Fermi-Ulam problem (the velocity and
207:{\displaystyle u_{n+1}=|u_{n}+U_{\mathrm {wall} }(\varphi _{n})|}
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Driven chaotic mesoscopic systems, dissipation and decoherence
1016:
E. Ott (1979). "Goodness of
Ergodic Adiabatic Invariants".
540:
with oscillating boundaries, The growth rate of energy in
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Over the years FUM became a prototype model for studying
908:"One-body dissipation and the super-viscidity of nuclei"
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871:D.H.E. Gross (1975). "Theory of nuclear friction".
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397:is the corresponding phase of the moving wall,
455:is the stochasticity parameter of the system.
729:https://doi.org/10.1016/S0375-9601(01)00670-3
8:
1334:Appl. Math. Sci. vol 38) (New York: Springer
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56:. The system consists of a particle that
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428:{\displaystyle U_{\mathrm {wall} }}
370:-th collision with the fixed wall,
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526:special theory of relativity
490:{\displaystyle (\varphi ,u)}
390:{\displaystyle \varphi _{n}}
21:Fermi–Ulam model (FUM)
1214:10.1103/physrevlett.85.1412
1116:10.1088/0305-4470/21/21/011
1073:10.1103/PhysRevLett.59.1173
1038:10.1103/PhysRevLett.42.1628
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73:Poincaré surface of section
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350:of the particle after the
462:invariant curves in the
108:{\displaystyle x=const.}
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58:collides elastically
40:FUM is a variant of
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1153:2000AnPhy.283..175C
1108:1988JPhA...21.4021W
1065:1987PhRvL..59.1173B
1030:1979PhRvL..42.1628O
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924:1978AnPhy.113..330B
885:1975NuPhA.240..472G
807:2008PhRvL.100a4103L
754:2008JPhA...41u2003G
685:2000JPhA...33.7973L
642:1988TMP....77.1110P
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508:non-linear dynamics
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1131:Annals of Physics
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464:phase space
460:KAM theorem
115:and writes
50:cosmic rays
1259:J. Phys. A
1137:(2): 175.
1096:J. Phys. A
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563:References
546:integrable
500:sinusoidal
912:Ann. Phys
798:0801.0641
658:120290250
615:250875392
538:billiards
476:φ
379:φ
248:φ
229:φ
188:φ
68:collide.
52:, namely
37:in 1961.
1344:Category
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595:Bibcode
542:chaotic
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319:where
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1311:link
1244:link
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