A class of nonrelativistic particle accelerators in which the majority of particles gain energy at an exponential rate is constructed. The class includes ergodic billiards with a piston that moves adiabatically and is removed adiabatically in a periodic fashion. The phenomenon is robust: deformations that keep the chaotic character of the billiard retain the exponential energy growth. The growth rate is found analytically and is, thus, controllable. Numerical simulations corroborate the analytic predictions with good precision. The acceleration mechanism has a natural thermodynamical interpretation and is applied to a hot dilute gas of repelling particles.

An unbounded energy growth of particles bouncing off two-dimensional (2D) smoothly oscillating polygons is observed. Notably, such billiards have zero Lyapunov exponents in the static case. For a special 2D polygon geometry--a rectangle with a vertically oscillating horizontal bar--we show that this energy growth is not only unbounded but also exponential in time. For the energy averaged over an ensemble of initial conditions, we derive an a priori expression for the rate of the exponential growth as a function of the geometry and the ensemble type. We demonstrate numerically that the ensemble averaged energy indeed grows exponentially, at a close to the analytically predicted rate-namely, the process is controllable.

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SignificanceDo partial energies in slow–fast Hamiltonian systems equilibrate? This is a long-standing problem related to the foundation of statistical mechanics. Altering the traditional ergodic assumption, we propose that nonergodicity in the fast subsystem leads to equilibration of the whole system. To show this principle, we introduce a set of mechanical toy models—the springy billiards—and describe stochastic processes corresponding to their adiabatic behavior. We expect that these models and this principle will play an important role in the quest to establish and study the underlying postulates of statistical mechanics, one of the long-standing scientific grails.

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