Mechanism of Fermi acceleration in dispersing billiards with time-dependent boundaries

Loskutov, A. Yu.; Ryabov, Alexey; Akinshin, L G

doi:10.1134/1.558939

Cited by 73 publications

(80 citation statements)

References 14 publications

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“…This inconsistency exists even for small a, so Karlis et al conclude that (69) is not a valid approximation of (68). We add that the same factor of two discrepancy can be observed between our quivering billiard expression for g 1 and the corresponding expressions obtained from the deterministic static wall maps given in [3,4,19,26]. In an early study of the FermiUlam model, Ref.…”

Section: Fixed Wall Simplificationsmentioning

confidence: 82%

“…An analogous static wall approximation can be constructed for higher dimensional billiards [19,26,29]. Like the quivering billiard, the static wall approximation eliminates the implicit equations for the time between collisions by holding the billiard boundary fixed.…”

Section: Fixed Wall Simplificationsmentioning

confidence: 99%

“…The two models differ because the static wall approximation assumes u(t b ) to be a well behaved function. It is common practice to consider stochastic versions of the maps (68) and (69), where u(t b ) is replaced by u(t b + ζ) for some random variable ζ [3,15,19,26,28,29]. The stochastic case simulates the effects of external noise on the system and allows one to average over ζ when determining ensemble averages, which often facilitates analytical calculations.…”

Section: Fixed Wall Simplificationsmentioning

confidence: 99%

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Universal energy diffusion in a quivering billiard

Demers

Jarzynski

2015

Phys. Rev. E

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We introduce and study a model of time-dependent billiard systems with billiard boundaries undergoing infinitesimal wiggling motions. The so-called quivering billiard is simple to simulate, straightforward to analyze, and is a faithful representation of time-dependent billiards in the limit of small boundary displacements. We assert that when a billiard's wall motion approaches the quivering motion, deterministic particle dynamics become inherently stochastic. Particle ensembles in a quivering billiard are shown to evolve to a universal energy distribution through an energy diffusion process, regardless of the billiard's shape or dimensionality, and as a consequence universally display Fermi acceleration. Our model resolves a known discrepancy between the one-dimensional Fermi-Ulam model and the simplified static wall approximation. We argue that the quivering limit is the true fixed wall limit of the Fermi-Ulam model.

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Section: Fixed Wall Simplificationsmentioning

confidence: 82%

Section: Fixed Wall Simplificationsmentioning

confidence: 99%

Section: Fixed Wall Simplificationsmentioning

confidence: 99%

See 1 more Smart Citation

Universal energy diffusion in a quivering billiard

Demers

Jarzynski

2015

Phys. Rev. E

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“…One may find in the literature examples of FA that may present transport distinct from the normal diffusion, as exponential [3][4][5][6], ballistic [7,8] or even slower growths [9,10]. Also, interesting FA applications can be found in research areas such as plasma physics [11][12][13], astrophysics [14,15], atom-optics [16,17], and especially billiard dynamics [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Transition from normal to ballistic diffusion in a one-dimensional impact system

et al. 2018

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We characterize a transition from normal to ballistic diffusion in a bouncing ball dynamics. The system is composed of a particle, or an ensemble of noninteracting particles, experiencing elastic collisions with a heavy and periodically moving wall under the influence of a constant gravitational field. The dynamics lead to a mixed phase space where chaotic orbits have a free path to move along the velocity axis, presenting a normal diffusion behavior. Depending on the control parameter, one can observe the presence of featured resonances, known as accelerator modes, that lead to a ballistic growth of velocity. Through statistical and numerical analysis of the velocity of the particle, we are able to characterize a transition between the two regimes, where transport properties were used to characterize the scenario of the ballistic regime. Also, in an analysis of the probability of an orbit to reach an accelerator mode as a function of the velocity, we observe a competition between the normal and ballistic transport in the midrange velocity.

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“…There are many results nowadays considering either static [7][8][9][10][11][12][13][14][15] and time dependent boundaries [16][17][18][19]. A phenomenon well known in time dependent boundary is the Fermi acceleration [20] as well as the so called Loskutov-Ryabov-Akinshin (LRA) conjecture [21,22]. Fermi acceleration [20] is a phenomenon where an ensemble of particles acquires unlimited energy from collisions with an infinitely heavy and moving wall.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics of a time-dependent and dissipative oval billiard: A heat transfer and billiard approach

et al. 2016

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We study some statistical properties for the behavior of the average squared velocity -hence the temperature -for an ensemble of classical particles moving in a billiard whose boundary is time dependent. We assume the collisions of the particles with the boundary of the billiard are inelastic leading the average squared velocity to reach a steady state dynamics for large enough time. The description of the stationary state is made by using two different approaches: (i) heat transfer motivated by the Fourier law and; (ii) billiard dynamics using either numerical simulations and theoretical description.

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Mechanism of Fermi acceleration in dispersing billiards with time-dependent boundaries

Cited by 73 publications

References 14 publications

Universal energy diffusion in a quivering billiard

Universal energy diffusion in a quivering billiard

Transition from normal to ballistic diffusion in a one-dimensional impact system

Thermodynamics of a time-dependent and dissipative oval billiard: A heat transfer and billiard approach

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