Spectral properties of the Laplacian and random matrix theories

Bohigas, O.; Giannoni, M.J.; Schmit, C.

doi:10.1051/jphyslet:0198400450210101500

Cited by 119 publications

(86 citation statements)

References 11 publications

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“…The Bohigas-Giannoni-Schmit conjecture [27] and later works suggest that for a chaotic system, it is appropriate to apply RMT arguments to understand its statistical behavior. Here, as mentioned already, one has to add the caveat -as long as ǫ is weak enough that C(ǫ; t) decays on a much longer time scale than the logtime.…”

Section: The Fidelitymentioning

confidence: 99%

A uniform approximation for the fidelity in chaotic systems

Cerruti

Tomsovic

2003

J. Phys. A: Math. Gen.

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Abstract. In quantum/wave systems with chaotic classical analogs, wavefunctions evolve in highly complex, yet deterministic ways. A slight perturbation of the system, though, will cause the evolution to diverge from its original behavior increasingly with time. This divergence can be measured by the fidelity, which is defined as the squared overlap of the two time evolved states. For chaotic systems, two main decay regimes of either Gaussian or exponential behavior have been identified depending on the strength of the perturbation. For perturbation strengths intermediate between the two regimes, the fidelity displays both forms of decay. By applying a complementary combination of random matrix and semiclassical theory, a uniform approximation can be derived that covers the full range of perturbation strengths. The time dependence is entirely fixed by the density of states and the so-called transition parameter, which can be related to the phase space volume of the system and the classical action diffusion constant, respectively. The accuracy of the approximations are illustrated with the standard map.

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Section: The Fidelitymentioning

confidence: 99%

A uniform approximation for the fidelity in chaotic systems

Cerruti

Tomsovic

2003

J. Phys. A: Math. Gen.

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“…[18][19][20][21][22] This result allows for a nonparametric description of the spacings distribution, i.e., a description that does not depend on the underlying random parameters. The degree of randomness can be quantified by the statistical overlap factor, which is defined as 34 …”

Section: B Spacing Statistics and The Goementioning

confidence: 99%

“…For the point load, the variance of the modal force is directly obtained from Eqs. (9) and (21). For the line load, Eqs.…”

Section: A Single Thin Plate Point and Line Loadingmentioning

confidence: 99%

“…Experimental 18-20 and numerical 21,22 studies have shown that the statistics of the local spacings between the eigenvalues saturate to those of the Gaussian orthogonal ensemble (GOE) matrix from random matrix theory. 23 This random matrix was first introduced in nuclear physics to represent a random Hamiltonian [24][25][26] and it bears little resemblance to the matrices that arise in the mathematical description of vibro-acoustic systems.…”

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confidence: 99%

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An efficient probabilistic approach to vibro-acoustic analysis based on the Gaussian orthogonal ensemble

Reynders

Legault

Langley

2014

The Journal of the Acoustical Society of America

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Vibro-acoustic analysis of complex systems at higher frequencies faces two challenges: how to compute the response without using an excessive number of degrees of freedom (DOFs), and how to quantify the uncertainty of the response due to small spatial variations in geometry, material properties, and boundary conditions, which have a wave scattering effect? In this study, a general method of analysis is presented that provides an answer to both questions while overcoming most limitations of statistical energy analysis. The fundamental idea is to numerically compute an artificial ensemble of realizations for the components of the built-up system that are highly sensitive to small random wave scatterers. This can be efficiently performed because their eigenvalue spacings and mode shapes conform to Gaussian orthogonal ensemble spacings and Gaussian random fields, respectively. The DOFs of the overall system are therefore limited to those of the deterministic components and the interface DOFs of the random components. The method is extensively validated by application to plate structures. Good agreement between the predicted response probability distributions and the results of detailed parametric probabilistic models is obtained, also for cases of low modal overlap, single point loading, and strong subsystem coupling.

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“…Subsequently, Dyson showed that the statistical predictions of RMT could be reproduced within a Brownian motion model, where Brownian motion of the energy levels results from a stochastic perturbation of some initial, possibly nonrandom, Hamiltonian [1]. Bohigas, Giannoni, and Schmit argued that the same universal behavior of the spectrum occurs generically even in single-particle systems, as long as the underlying classical dynamics is chaotic [2].…”

Section: Introductionmentioning

confidence: 99%

Brownian motion model of quantization ambiguity and universality in chaotic systems

Kaplan

2005

Phys. Rev. E

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We examine spectral equilibration of quantum chaotic spectra to universal statistics, in the context of the Brownian motion model. Two competing time scales, proportional and inversely proportional to the classical relaxation time, jointly govern the equilibration process. Multiplicity of quantum systems having the same semiclassical limit is not sufficient to obtain equilibration of any spectral modes in two-dimensional systems, while in three-dimensional systems equilibration for some spectral modes is possible if the classical relaxation rate is slow. Connections are made with upper bounds on semiclassical accuracy and with fidelity decay in the presence of a weak perturbation.

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Spectral properties of the Laplacian and random matrix theories

Cited by 119 publications

References 11 publications

A uniform approximation for the fidelity in chaotic systems

A uniform approximation for the fidelity in chaotic systems

An efficient probabilistic approach to vibro-acoustic analysis based on the Gaussian orthogonal ensemble

Brownian motion model of quantization ambiguity and universality in chaotic systems

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