Universal energy diffusion in a quivering billiard

Abstract

We introduce and study a model of time-dependent billiard systems with billiard boundaries undergoing infinitesimal wiggling motions. The so-called quivering billiard is simple to simulate, straightforward to analyze, and is a faithful representation of time-dependent billiards in the limit of small boundary displacements. We assert that when a billiard's wall motion approaches the quivering motion, deterministic particle dynamics become inherently stochastic. Particle ensembles in a quivering billiard are shown to evolve to a universal energy distribution through an energy diffusion process, regardless of the billiard's shape or dimensionality, and as a consequence universally display Fermi acceleration. Our model resolves a known discrepancy between the one-dimensional Fermi-Ulam model and the simplified static wall approximation. We argue that the quivering limit is the true fixed wall limit of the Fermi-Ulam model.

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

“…valid for Gibbs ensembles or for individual eigenstates in very large systems. This more general fluctuation-dissipation relation for individual eigenstates (224) still connects the noise and the dissipative response but in a more complicated way.…”

“…If this is the case, then one can consider a continuous driving protocol instead of repeated quenches and all the results will be the same. Such a setup was analyzed by Jarzynski [213] followed by other works [193,[220][221][222][223][224]]. An interesting and nontrivial result that emerges from this analysis is a nonequilibrium exponential velocity distribution (to be contrasted with the Gaussian Maxwell distribution).…”

“…Defining τ = αt = tC d /[2(d + 1)], we obtain the final result for the asymptotic energy distribution P (E, τ ) (see also Refs. [223,224]):…”

Energy diffusion and absorption in chaotic systems with rapid periodic driving

“…We can rewrite this expression in terms of γ E0 (x), the differential average collision rate for collisions at the lo-cation x. γ E0 (x) is obtained by integrating ρ E0 (x, v) v • n(x) over all v such that v • n(x) > 0. This is another spherical integral; the result is given by (18). Comparing (18) and (A10), we obtain (16).…”

Wake turbulence observed behind an upstream “extra” particle in a complex (dusty) plasma

“…Dynamically, the highly-collisional wake observed in the given experiments is equivalent to a time-dependent billiard system, and should be treated correspondingly [45]. The description of the wake in our case could be significantly simplified, though.…”