Slow relaxation in weakly open rational polygons

Abstract

The interplay between the regular (piecewise-linear) and irregular (vertex-angle) boundary effects in nonintegrable rational polygonal billiards (of m equal sides) is discussed. Decay dynamics in polygons (of perimeter P(m) and small opening Delta) is analyzed through the late-time survival probability S(m) approximately equal t(-delta). Two distinct slow relaxation channels are established. The primary universal channel exhibits relaxation of regular sliding orbits, with delta=1. The secondary channel is given by delta>1 and becomes open when m>P(m)/Delta. It originates from vertex order-disorder dual effects and is due to relaxation of chaoticlike excitations.

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).…”

“…Also, the experimental observations was elaborated for the large opening ∆ = 0.20R, when β = 32 were derived in this case (see Fig. 3 in [21]). We examine that the upper limit for the α-channel-observation window shows its sensitivity to the opening width.…”

“…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].…”

“…Similarly, if one ignores the vertex-splitting (irregular) effects a given m-gon, the particle-collision function, corresponding to the regular orbits, can be obtained through the model (equivalent-side) approximation [21], namely…”