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We study the energy dynamics of a particle in a billiard subject to a rapidperiodic drive. In the regime of large driving frequencies $\omega$, we findthat the particle's energy evolves diffusively, which suggests that theparticle's energy distribution $\eta (E,t)$ satisfies a Fokker-Planck equation.We calculate the rates of energy absorption and diffusion associated with thisequation, finding that these rates are proportional to $\omega^{-2}$ for large$\omega$. Our analysis suggests three phases of energy evolution:Prethermalization on short timescales, then slow energy absorption inaccordance with the Fokker-Planck equation, and finally a breakdown of therapid driving assumption for large energies and high particle speeds. We alsopresent numerical simulations of the evolution of a rapidly driven billiardparticle, which corroborate our theoretical results.

Energy diffusion and prethermalization in chaotic billiards under rapid periodic driving