Wave chaos in the stadium: Statistical properties of short-wave solutions of the Helmholtz equation

McDonald, Steven W.; Kaufman, Allan N.

doi:10.1103/physreva.37.3067

Cited by 274 publications

(177 citation statements)

References 67 publications

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“…For two-degree-of-freedom systems, c(r) can be understood as being given approximately by a Bessel function. For distances |r − r ′ | short compared to the system size, and in the absence of effects related to classical dynamics [8,9,13], this is roughly observed in numerical [7,14,15] and experimental [16] studies. Our interest here is in statistical properties of eigenfunctions going beyond local quantities such as c(r).…”

mentioning

confidence: 73%

Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

2008

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New fluctuation properties arise in problems where both spatial integration and energy summation are necessary ingredients. The quintessential example is given by the short-range approximation to the first order ground state contribution of the residual Coulomb interaction. The dominant features come from the region near the boundary where there is an interplay between Friedel oscillations and fluctuations in the eigenstates. Quite naturally, the fluctuation scale is significantly enhanced for Neumann boundary conditions as compared to Dirichlet. Elements missing from random plane wave modeling of chaotic eigenstates lead surprisingly to significant errors, which can be corrected within a purely semiclassical approach. The characterization of quantum systems with any of a variety of underlying classical dynamics, ranging from diffusive to chaotic to regular, has often demonstrated that the study of their statistical properties is of primary importance. Spectral fluctuations are a principle example as they gave the first support to one of the main results linking classical chaos and random matrix theory [1], the Bohigas-Giannoni-Schmit conjecture [2,3]. Needless to say, the statistical properties of eigenfunctions are also a subject of paramount interest [4,5,6,7,8,9,10,11,12].For chaotic systems, a widely accepted starting point for the treatment of eigenfunction fluctuations locally, such as the amplitude distribution or the two-point correlation function c(|r − r ′ |) = ψ(r)ψ(r ′ ) of a given eigenfunction ψ, is a modeling in terms of a random superposition of plane waves (RPW) [4,5]. For two-degree-of-freedom systems, c(r) can be understood as being given approximately by a Bessel function. For distances |r − r ′ | short compared to the system size, and in the absence of effects related to classical dynamics [8,9,13], this is roughly observed in numerical [7,14,15] and experimental [16] studies. Our interest here is in statistical properties of eigenfunctions going beyond local quantities such as c(r).One motivation for the introduction of these new statistical measures is to study the interplay between interferences and interactions in mesoscopic systems. For typical electronic densities, the screening length is close to the Fermi wavelength λ F , and the screened Coulomb interaction can be approximated by the short range expression V sc (r − r ′ ) = (2ν) −1 F a 0 δ(r − r ′ ) with 2ν the mean local density of states, including spin degeneracy, (ν = m/2π 2 for d = 2) and To this level of approximation, the first order ground state energy contribution of the residual interactions can be expressed in the formwith N σ the unperturbed ground state density of particles with spin σ. From this expression, it is seen that the increase of interaction energy associated with the addition of an extra electron is related toand the understanding that E i < E + i < E i+1 . Our goal in this letter is to study the fluctuation properties of the S i , concentrating on the case of two dimensional billiards with either Dirichlet ...

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confidence: 73%

Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

2008

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“…Density plot of |ψ n (x, y)| 2 for a series of bouncing ball modes in the stadium billiard, with n = 320 (13,1), n = 321 (13,2), n = 325 (13,3), n = 329 (13,4), n = 333 (13,5) and n = 339 (13,6). The numbers in brackets denote the adiabatic quantum numbers (l, k) being the number of modes in the x-and y-direction, respectively.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Already in [2] bouncing ball-like states were discussed and then were observed in the stadium billiard [3,4]. A method of constructing these states approximately was given in [5].…”

Section: Introductionmentioning

confidence: 99%

On the number of bouncing ball modes in billiards

Bäcker

Schubert

Stifter

1997

J. Phys. A: Math. Gen.

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Abstract. We study the number of bouncing ball modes N bb (E) in a class of two-dimensional quantized billiards with two parallel walls. Using an adiabatic approximation we show that asymptotically N bb (E) ∼ αE δ for E → ∞, where δ ∈] 1 2 , 1[ depends on the shape of the billiard boundary. In particular for the class of two-dimensional Sinai billiards, which are chaotic, one can get arbitrarily close (from below) to δ = 1, which corresponds to the leading term in Weyl's law for the mean behaviour of the counting function of eigenstates. This result shows that one can come arbitrarily close to violating quantum ergodicity. We compare the theoretical results with the numerically determined counting function N bb (E) for the stadium billiard and the cosine billiard and find good agreement.

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“…Such discrepancies are not new. Already in the disseminating paper by McDonald and Kaufman [3] Gaussian distributions for ψ were observed for chaotic wave functions exclusively. It is obvious that the random-superposition-of-plane-waves approach cannot work for bouncing-ball and scarred wave functions such as the one shown in Fig.…”

Section: Intensity Distributionsmentioning

confidence: 99%

Current and vortex statistics in microwave billiards

Barth

Stöckmann

2002

Phys. Rev. E

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Using the one-to-one correspondence between the Poynting vector in a microwave billiard and the probability current density in the corresponding quantum system probability densities and currents were studied in a microwave billiard with a ferrite insert as well as in an open billiard. Distribution functions were obtained for probability densities, currents and vorticities. In addition the vortex pair correlation function could be extracted. For all studied quantities a complete agreement with the predictions from the random-superposition-of-plane-waves approach was obtained.

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Wave chaos in the stadium: Statistical properties of short-wave solutions of the Helmholtz equation

Cited by 274 publications

References 67 publications

Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

On the number of bouncing ball modes in billiards

Current and vortex statistics in microwave billiards

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