Unitarity and irreversibility in chaotic systems

Hasegawa, Hiroshi; Saphir, William

doi:10.1103/physreva.46.7401

Cited by 100 publications

(88 citation statements)

References 24 publications

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“…This becomes increasingly difficult as the dynamics gains complexity. Especially for the large class of systems with a mixed phase space an effective approximation scheme is needed that is free of restrictions of previous investigations [10,11], such as hyperbolicity, one-dimensional (quasi-) phase space, or isolation of the phase-space regions causing intermittency.…”

Section: Introductionmentioning

confidence: 99%

Resonances of the Frobenius-Perron operator for a Hamiltonian map with a mixed phase space

Weber

Haake

Braun

et al. 2001

J. Phys. A: Math. Gen.

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Resonances of the (Frobenius-Perron) evolution operator P for phase-space densities have recently attracted considerable attention, in the context of interrelations between classical and quantum dynamics. We determine these resonances as well as eigenvalues of P for Hamiltonian systems with a mixed phase space, by truncating P to finite size in a Hilbert space of phase-space functions and then diagonalizing. The corresponding eigenfunctions are localized on unstable manifolds of hyperbolic periodic orbits for resonances and on islands of regular motion for eigenvalues. Using information drawn from the eigenfunctions we reproduce the resonances found by diagonalization through a variant of the cycle expansion of periodic-orbit theory and as rates of correlation decay.

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Section: Introductionmentioning

confidence: 99%

Resonances of the Frobenius-Perron operator for a Hamiltonian map with a mixed phase space

Weber

Haake

Braun

et al. 2001

J. Phys. A: Math. Gen.

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“…While for an open system, the escape rate γ 0 is nonvanishing, for closed system, γ 0 = 0, and the corresponding eigenstate is just the equilibrium state. The RP resonances are essential ingredients of a description of irreversibility and thermodynamics based upon microscopic chaos [40,9,41]. However these resonances have remained mathematical objects.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

Quantum Correlations and Classical Resonances in an Open Chaotic System

Pance

Pradhan

et al. 2001

Quantum Chaos Y2K

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We show that the autocorrelation of quantum spectra of an open chaotic system is well described by the classical RuellePollicott resonances of the associated chaotic strange repeller. This correspondence is demonstrated utilizing microwave experiments on 2-D n-disk billiard geometries, by determination of the wave-vector autocorrelation C(κ) from the experimental quantum spectra S21(k). The correspondence is also established via "numerical experiments" that simulate S21(k) and C(κ) using periodic orbit calculations of the quantum and classical resonances. Semiclassical arguments that relate quantum and classical correlation functions in terms of fluctuations of the density of states and correlations of particle density are also examined and support the experimental results. The results establish a correspondence between quantum spectral correlations and classical decay modes in an open systems.

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“…This shows that <<pj should be distributions which have a meaning with a suitable choice of test functions [7,9]. It is remarkable that even in this simple classical problem we have to introduce a "rigged Hilbert space" formalism to include the Lyapounoff time in the spectrum.…”

Section: (22)mentioning

confidence: 99%

“…More generally it has been shown recently that the right eigenfunctions of U are the Bernoulli polynomials ß"(x), corresponding to the eigenvalues (|) n [7,8].…”

Section: Classical Dynamicsmentioning

confidence: 99%

“…We may even introduce a dynamic analogue of Boltzmann's JC-function. This is given by the operator [24] jr = |0i><0il, (4)(5)(6)(7)(8)(9)(10)(11)(12) which in conjunction with "test functions" defined in the Hilbert space satisfies the inequalities <«P|Jf |f>>0, | f> <0. (4.13) df Similar results have been obtained by Sudarshan, Chiu and Gorini [29] in their important paper "Decaying states as complex energy eigenvalues in generalized quantum mechanics".…”

Section: Quantum Theory Of Non-integrable Systems: Friedrichs Modelmentioning

confidence: 99%

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From Poincare's Divergences to Quantum Mechanics with Broken Time Symmetry

Prigogine

1997

Zeitschrift Für Naturforschung A

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We discuss the spectral property of unstable dynamical systems in both classical and quantum mechanics. An important class of unstable dynamical systems corresponds to the Large Poincare Systems (LPS). Conventional perturbation technique leads then to divergences. We introduce methods for the elimination of Poincare divergences to obtain a solution of the spectral problem analytic in the coupling constant. To do so, we have to enlarge the class of permissible transformations, to include non-unitary transformations as well as to extend the Hilbert space. A simple example refers to the Friedrichs model, which was studied independently by George Sudarshan and his co-workers. However, our main interest is the irreducible representations in the Liouville space. In these representations the central quantity is the density matrix, and the eigenfunctions of the Liouville operator cannot be expressed in terms of the wave functions. We suggest that this situation corresponds to quantum chaos. Indeed, classical chaos does not mean that Newton's equation becomes "wrong" but that trajectories loose their operational meaning. Similarly, whenever we have an irreducible representation in the Liouville space this means that the wave function description looses its operational meaning. Additional statistical features appear. A simple example corresponds to persistent interactions in the scattering problem which cannot be treated in the frame of usual S-matrix theory.

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Unitarity and irreversibility in chaotic systems

Cited by 100 publications

References 24 publications

Resonances of the Frobenius-Perron operator for a Hamiltonian map with a mixed phase space

Resonances of the Frobenius-Perron operator for a Hamiltonian map with a mixed phase space

Quantum Correlations and Classical Resonances in an Open Chaotic System

From Poincare's Divergences to Quantum Mechanics with Broken Time Symmetry

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