Semiquantum chaos and the large N expansion

Cooper, Fred; Dawson, John F.; Habib, Salman; Kluger, Yuval; Meredith, Dawn C.; Shepard, Harvey K.

doi:10.1016/0167-2789(94)00251-k

Cited by 11 publications

(11 citation statements)

References 21 publications

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“…A different but equivalent set of canonical variables was discussed in Ref. [7]. Hence by the simple change of notation in (3.2) and (3.4) we have recognized that the large N equations for the quantum anharmonic oscillator with classical potential…”

Section: The Effective Hamiltonian and Density Matrixmentioning

confidence: 99%

Nonequilibrium dynamics of symmetry breaking inλΦ4theory

et al. 1997

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The time evolution of O(N ) symmetric λΦ 4 scalar field theory is studied in the large N limit. In this limit the Φ mean field and two-point correlation function ΦΦ evolve together as a self-consistent closed Hamiltonian system, characterized by a Gaussian density matrix. The static part of the effective Hamiltonian defines the True Effective Potential U ef f for configurations far from thermal equilibrium. Numerically solving the time evolution equations for energy densities corresponding to a quench in the unstable spinodal region, we find results quite different from what might be inferred from the equilibrium free energy "effective" potential F . Typical time evolutions show effectively irreversible energy flow from the coherent mean fields to the quantum fluctuating modes, due to the creation of massless Goldstone bosons near threshold. The plasma frequency and collisionless damping rate of the mean fields are calculated in terms of the particle number density by a linear response analysis and compared with the numerical results. Dephasing of the fluctuations leads also to the growth of an effective entropy and the transition from quantum to classical behavior of the ensemble. In addition to casting some light on fundamental issues of nonequilibrium quantum statistical mechanics, the general framework presented in this work may be applied to a study of the dynamics of second order phase transitions in a wide variety of LandauGinsburg systems described by a scalar order parameter.

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Section: The Effective Hamiltonian and Density Matrixmentioning

confidence: 99%

Nonequilibrium dynamics of symmetry breaking inλΦ4theory

et al. 1997

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“…In this sort of approximation of a quantum system coupled with a semiclassical degree of freedom such as a coherent electric or gravitational field, the approximate dynamics of the quantum system can become chaotic. This was first described by us, and termed "semiquantum chaos" [1,2]. A closely related result having the same cause is "semiquantal chaos" [3] which occurs in the time-dependent Gaussian approximation for the dynamics of quantum systems.…”

Section: Introductionmentioning

confidence: 92%

“…The first keeps Gaussian correlations (Hartree approximation) for both oscillators, while the second (large N approximation) ignores fluctuations in the A oscillator. The second approximation has been derived previously from a path integral approach [2] by making N copies of the x oscillator and then taking the large N limit.…”

Section: Hartree Approximation and The Large N Limitmentioning

confidence: 99%

“…(This system was discussed in detail in Refs. [1,2].) It is also perfectly clear that the large N limit is equivalent to treating the A oscillator classically (i.e., ignoring the quantum fluctuations about the mean value of A).…”

Section: Hartree Approximation and The Large N Limitmentioning

confidence: 99%

“…We will then focus on exactly the same model system treated in Refs. [1,2], namely a system of two coupled oscillators described by the Lagrangian:…”

Section: Introductionmentioning

confidence: 99%

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Chaos in time-dependent variational approximations to quantum dynamics

et al. 1998

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Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first note that any variational approximation to the dynamics of a quantum system based on the Dirac action principle leads to a classical Hamiltonian dynamics for the variational parameters. Since this Hamiltonian is generically nonlinear and nonintegrable, the dynamics thus generated can be chaotic, in distinction to the exact quantum evolution. We then restrict our attention to a system of two biquadratically coupled quantum oscillators and study two variational schemes, the leading order large-N ͑four canonical variables͒ and Hartree ͑six canonical variables͒ approximations. The chaos seen in the approximate dynamics is an artifact of the approximations: this is demonstrated by the fact that its onset occurs on the same characteristic time scale as the breakdown of the approximations when compared to numerical solutions of the time-dependent Schrödinger equation.

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Parallel algorithm with spectral convergence for nonlinear integro-differential equations

Mihaila

Shaw

2002

J. Phys. A: Math. Gen.

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We discuss a numerical algorithm for solving nonlinear integro-differential equations, and illustrate our findings for the particular case of Volterra type equations. The algorithm combines a perturbation approach meant to render a linearized version of the problem and a spectral method where unknown functions are expanded in terms of Chebyshev polynomials (El-gendi's method). This approach is shown to be suitable for the calculation of two-point Green functions required in next to leading order studies of time-dependent quantum field theory.

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Semiquantum chaos and the large N expansion

Cited by 11 publications

References 21 publications

Nonequilibrium dynamics of symmetry breaking inλΦ4theory

Nonequilibrium dynamics of symmetry breaking inλΦ4theory

Chaos in time-dependent variational approximations to quantum dynamics

Parallel algorithm with spectral convergence for nonlinear integro-differential equations

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