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Fermi acceleration in time-dependent billiards: theory of the velocity diffusion in conformally breathing fully chaotic billiards
Batistić, Benjamin ; Robnik, Marko, 1954-
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 ...(periodic oscillation of the boundary which preserves the shape of the billiard at all times), which are fully chaotic as static (frozen) billiards, and we show that for large velocities around v0 and for not too long times, we observe just normal diffusion of the velocity as a function of the physical (continuous) time, around v0. However, the diffusion is not homogeneous, as the diffusion constant D depends on v0 as a power law D å 1/v3 0 . Taking this into account, we show that to the leading order the average velocity v(n) as a function of the number of collisions n obeys a power law v å n1/6; thus, the Fermi acceleration exponent is fÀ = 1/6, which is in excellent agreement with the numerical calculations of the fully chaotic oval billiard, the Sinai billiard and the cardioid billiard. The error of the velocity estimates is of the order 1/v2. Thus, the higher the velocity, the better our analytic approximation. Moreover, we derive the underlying universal equation of the velocity dynamics of the time-dependent conformally breathing billiards, correct up to and including the order 1/v in the regime of the large velocity of the particle v. This universal equation does not depend on the dynamical properties of the system (integrability, ergodicity, chaoticity). We present the results of the numerical simulations for three billiards in complete agreement with the theory. We believe that this is a first step towards theoretical understanding of the power law growth and the Fermi acceleration exponents in 2D billiards, although our theory is so far specialized to the conformally breathing fully chaotic billiards.
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