From ballistic to Brownian motion through enhanced diffusion in vertex-splitting polygonal and disk-dispersing Sinai billiards

Abstract

Boundary-collision orbit statistics in vertex-splitting rational polygonal and disk-scattering billiards is studied using deterministic and stochastic schemes. On increasing the number of vertices in pseudointegrable polygons, the diffusion exponent, deduced from the mean-square orbit displacement, exhibits a crossover from a ballistic to a superdiffusive regime, characteristic of chaotic Sinai billiards.

On Chaotic Dynamics in Rational Polygonal Billiards

“…was also found [12,21], in the regular-orbit approximation. Here ψ m = ϕ cm and ψ m = 0, for, respectively, the even m-gon and odd m-gon cases, discussed in equation (9).…”

“…Being related to the zeromeasure singularities in phase space, these effects violate the integrability of polygons [1], as well as the classical-to-quantum correspondence principle [12]. The latter finding is due to the memory of vertex-splitting effects, which do not disappear in the billiard dynamics when m → ∞, and thereby forbid the interchange of temporal (t → ∞) and the spatial (m → ∞) limits.…”

“…We have communicated [12] that the chaotic disk-dispersing SBs and non-chaotic vertex-splitting polygonal billiards display a similar behavior attributed to the enhanced diffusion dynamics. In polygons, this late-time dynamics is associated with the singular arc-splitting effects, produced by singularities of orbits provoked by the vertices [1].…”

“…Consequently, the orbit-set classification of the late-time living orbits in such non-chaotic billiards can be introduced through the collision angle ϕ = [0, π/2], estimated from the normal to the boundary wall [10,20]. This motion integral provides the basis for our deterministic approach to billiard dynamics proposed in [12] and then developed for the square billiard [6] and polygons [21].…”

“…The stochastic approach proposed for analysis of these effects is based on exploration of the generalized diffusion equation, which manifests the dynamic relaxation of non-Markovian time evolution [13]. This fundamental model-independent equation, which generalizes the Brownian diffusion, is given for the variance of particle displacements ∆ 2 r(t) realized during time t. In application to m-gons, this equation was introduced through the variance of the number of random collisions [12], namely…”