Fermi acceleration and adiabatic invariants for non-autonomous billiards.

@article{Gelfreich2012FermiAA,
  title={Fermi acceleration and adiabatic invariants for non-autonomous billiards.},
  author={V. Gelfreich and Vered Rom-Kedar and Dmitry Turaev},
  journal={Chaos},
  year={2012},
  volume={22 3},
  pages={
          033116
        }
}
Recent results concerned with the energy growth of particles inside a container with slowly moving walls are summarized, augmented, and discussed. For breathing bounded domains with smooth boundaries, it is proved that for all initial conditions the acceleration is at most exponential. Anosov-Kasuga averaging theory is reviewed in the application to the non-autonomous billiards, and the results are corroborated by numerical simulations. A stochastic description is proposed which implies that… 

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References

SHOWING 1-10 OF 62 REFERENCES

Robust exponential acceleration in time-dependent billiards.

A class of nonrelativistic particle accelerators in which the majority of particles gain energy at an exponential rate is constructed and is robust: deformations that keep the chaotic character of the billiard retain the exponential energy growth.

The presence and lack of Fermi acceleration in nonintegrable billiards

The unlimited energy growth (Fermi acceleration) of a classical particle moving in a billiard with a parameter-dependent boundary oscillating in time is numerically studied. The shape of the boundary

Tunable fermi acceleration in the driven elliptical billiard.

The existence of Fermi acceleration is shown thereby refuting the established assumption that smoothly driven billiards whose static counterparts are integrable do not exhibit acceleration dynamics and it is possible to tune the acceleration law in a straightforwardly controllable manner.

Stochastic and Adiabatic Behavior of Particles Accelerated by Periodic Forces

The mechanism by which periodic nonrandom forces lead to stochastic acceleration of particles is examined. Two examples considered are (i) the Fermi problem of a ball bouncing between a fixed and an

Fermi acceleration in non-autonomous billiards

Fermi acceleration can be modelled by a classical particle moving inside a time-dependent domain and elastically reflecting from its boundary. In this paper, we describe how the results from the

Exponential energy growth in a Fermi accelerator.

It is demonstrated numerically that the ensemble averaged energy indeed grows exponentially, at a close to the analytically predicted rate-namely, the process is controllable.

Fermi acceleration in time-dependent billiards: theory of the velocity diffusion in conformally breathing fully chaotic billiards

We study aspects of the Fermi acceleration (the unbounded growth of the energy) in a certain class of time-dependent 2D billiards. Specifically, we look at the conformally breathing billiards

Phase-space composition of driven elliptical billiards and its impact on Fermi acceleration.

This work considers three different driving modes of the elliptical billiard and performs a comprehensive analysis of the corresponding four-dimensional phase space and shows that the stickiness properties, which eventually determine the diffusion, are intimately connected with this change in the composition of the phase space with respect to velocity.

Energy growth rate in smoothly oscillating billiards.

  • Kushal Shah
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2011
It is shown that the energy growth rate in oscillating pseudo-integrable billiards could be exponential in time, important for applications since slower than exponential-in-time energy growth can be annihilated by dissipation.

Hyperacceleration in a stochastic Fermi-Ulam model.

An improved approximative map is introduced, which takes into account the effect of the wall displacement, and in addition allows the analytical estimation of the long term behavior of the particle mean velocity as well as the corresponding probability distribution, in complete agreement with the numerical results of the exact dynamics.
...