Sinai billiard

Even for the resonant transmission through the Sinai billiard, computations show that many eigenfunctions contribute to the scattering wave function as shown in fig. 1. An assumption of a complex RGF for the scattering function (9) means that the joint probability density has the form... 

Figure 1. The coefficients cn in the expansion (8) for the resonant transmission through the Sinai billiard with numerical sizes 500 x 500, R = 50 with energy e = 10.425. 

Figure 2. Statistics of current for the transmission through the Sinai billiard for T 0. The upper left panel shows the computed distribution for p = 2 together with the Porter-Thomas distribution P(p) (solid curve). In the inset in the same panel the computed wave function statistics f(p) for the real part of ip is compared with a random Gaussian distribution (solid curve). 

Figure 4 Distributions for separations between the nearest distances nearest NPs, saddle points, NPs with the same (++) and opposite winding numbers (+-) in a chaotic Sinai billiard. The radial distribution of nearest distances for completely random points (26) is shown by the dashed curve in (a). The corresponding distribution for the Berry model function for a chaotic state (2) and random superposition of 16 eigen functions for a rectangular box with the same size and energy are shown by dots and thin curves, respectively.

Figure 2. Nearest-neighbor spacing distributions of eigenvalues for the Sinai billiard with the Wigner surmise compared to the Poisson distribution. The histogram comprises about 1000 consecutive eigenvalues. Taken from Ref. (Bohigas, Giannoni and Schmit, 1984).

Very accurate results were obtained for the classically chaotic Sinai billiard by Bohigas, Giannoni, and Schmit (see Fig. 2) which led them to the important conclusion (Bohigas, Giannoni and Schmit, 1984) Spectra of time-reversal invariant systems whose classical analogues are K systems show the same fluctuation properties as predicted by the Gaussian orthogonal ensemble (GOE) of random-matrix theory... 

Primack, H. and Smilansky, U. (1995). Quantization of the three-dimensional Sinai billiard, Phys. Rev. Lett. 74, 4831-4834. 

Figure 3. The complexity of nodal lines, nodal points and saddles for the transmission through chaotic (Sinai) (left) and regular billiard (right). 

The periodic hard-disk Lorentz gas is a two-dimensional billiard in which a point particle undergoes elastic collisions on hard disks which are fixed in the plane in the form of a spatially periodic lattice. Bunimovich and Sinai have proved that... 

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