title: Fermi acceleration in chaotic shape-preserving billiards.
abstract: We study theoretically and numerically the velocity dynamics of fully chaotic time-dependent shape-preserving billiards. The average velocity of an ensemble of initial conditions generally asymptotically follows the power law 〈v〉 = n(β) with respect to the number of collisions n. If a shape of a fully chaotic time-dependent billiard is not preserved it is well known that the acceleration exponent is β = 1/2. We show, on the other hand, that if a shape of a fully chaotic time-dependent billiard is preserved then there are only three possible values of β depending solely on the rotational properties of the billiard. In a special case in which the only transformation is a uniform rotation there is no acceleration, β = 0. Excluding this special case, we show that if a time-dependent transformation of a billiard is such that the angular momentum of the billiard is preserved then β = 1/6, while β = 1/4 otherwise. Our theory is centered around the detailed study of the energy fluctuations in the adiabatic limit. We show that three quantities, two scalars and one tensor, completely determine the energy fluctuations of the billiard for arbitrary time-dependent shape-preserving transformations. Finally, we provide several interesting numerical examples, all in a perfect agreement with the theory.
authors.affiliation: Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia.
authors.forename: Benjamin
authors.initials: B
authors.lastname: Batistić