Leaky Fermi accelerators

A Fermi accelerator is a billiard with oscillating walls. A leaky accelerator interacts with an environment of an ideal gas at equilibrium by exchange of particles through a small hole on its boundary. Such interaction may heat the gas: we estimate the net energy flow through the hole under the assumption that the particles inside the billiard do not collide with each other and remain in the accelerator for a sufficiently long time. The heat production is found to depend strongly on the type of Fermi accelerator. An ergodic accelerator, i.e., one that has a single ergodic component, produces a weaker energy flow than a multicomponent accelerator. Specifically, in the ergodic case the energy gain is independent of the hole size, whereas in the multicomponent case the energy flow may be significantly increased by shrinking the hole size.

Figure 1

(a) Sinai accelerator, (b) divided Sinai accelerator, (c) mushroom accelerator, and (d) stadium accelerator. See the simulation section (Sec. 3) for geometric specifications and dynamical properties.

Figure 2

The distribution and the average number of collisions; see Eq. (3). (a) Dependence on the hole size. Here, the initial energy is $E_{\text{in}}=9000$ for the Sinai and divided Sinai accelerators, and $E_{\text{in}}=1250$ for the mushroom and stadium accelerators. (b) No dependence on the initial particles' energy. Here $h=0.0005$ for Sinai and divided Sinai accelerators, and $h=0.00033$ for mushroom and stadium accelerators. (c) The exponential distribution of the exit probabilities for the stadium accelerator. Here $E_{\text{in}}=1250$ and $h=0.00033$.

Figure 3

Averaged energy gain dependence on the hole size. The energy gain increases linearly with $1/h$ for multicomponent billiards [see Eq. (16)], and it is independent of the hole size for the ergodic billiards (for sufficiently small holes); see Eq. (7). Here, the initial energy is $E_{\text{in}}=9000$ for the Sinai and divided Sinai accelerators, and $E_{\text{in}}=1250$ for the mushroom and stadium accelerators.

Figure 4

Dependence of the average energy gain on the initial speed: linear growth with $v_{\text{in}}$ for multicomponent billiards [Eq. (14)] and slow growth with $v_{\text{in}}$ for the ergodic case, possibly due to order $h$ corrections to Eq. (6). Here $h=0.0005$ for Sinai and divided Sinai accelerators, and $h=0.00033$ for mushroom and stadium accelerators.

Figure 5

The speed bias vs initial speed. Here $h=0.0005$ for Sinai and divided Sinai, and $h=0.00033$ for mushroom and stadium.