Generalized Lyapunov exponents of the random harmonic oscillator: Cumulant expansion approach

Vallejos, Raúl O.; Anteneodo, Celia

doi:10.1103/physreve.85.021124

Cited by 10 publications

(7 citation statements)

References 64 publications

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“…As correctly pointed out by Vallejos and Anteneodo in Ref. [49] and also by Politi [50], Eq. (23) does not directly estimate the Lyapunov exponent λ but rather a generalized Lyapunov exponent commonly referred to as λ 2 .…”

Section: B Geometry and Chaosmentioning

confidence: 58%

Geometry of the energy landscape of the self-gravitating ring

Monechi

Casetti

2012

Phys. Rev. E

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We study the global geometry of the energy landscape of a simple model of a self-gravitating system, the self-gravitating ring (SGR). This is done by endowing the configuration space with a metric such that the dynamical trajectories are identified with geodesics. The average curvature and curvature fluctuations of the energy landscape are computed by means of Monte Carlo simulations and, when possible, of a mean-field method, showing that these global geometric quantities provide a clear geometric characterization of the collapse phase transition occurring in the SGR as the transition from a flat landscape at high energies to a landscape with mainly positive but fluctuating curvature in the collapsed phase. Moreover, curvature fluctuations show a maximum in correspondence with the energy of a possible further transition, occurring at lower energies than the collapsed one, whose existence had been previously conjectured on the basis of a local analysis of the energy landscape and whose effect on the usual thermodynamic quantities, if any, is extremely weak. We also estimate the largest Lyapunov exponent λ of the SGR using the geometric observables. The geometric estimate always gives the correct order of magnitude of λ and is also quantitatively correct at small energy densities and, in the limit N→∞, in the whole homogeneous phase.

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Section: B Geometry and Chaosmentioning

confidence: 58%

Geometry of the energy landscape of the self-gravitating ring

Monechi

Casetti

2012

Phys. Rev. E

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“…Finally, we note that the quantitative disagreement between theoretical estimates and numerical measurements of λ when σ k 0 may also have a different origin. As correctly pointed out by Vallejos and Anteneodo in [48] and also by Politi [49], Eq. ( 23) does not directly estimate the Lyapunov exponent λ but rather a generalized Lyapunov exponent commonly referred to as λ 2 .…”

Section: B Geometry and Chaosmentioning

confidence: 64%

Geometry of the energy landscape of the self-gravitating ring

Monechi,

Casetti

2012

Preprint

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We study the global geometry of the energy landscape of a simple model of a self-gravitating system, the self-gravitating ring (SGR). This is done by endowing the configuration space with a metric such that the dynamical trajectories are identified with geodesics. The average curvature and curvature fluctuations of the energy landscape are computed by means of Monte Carlo simulations and, when possible, of a mean-field method, showing that these global geometric quantities provide a clear geometric characterization of the collapse phase transition occurring in the SGR as the transition from a flat landscape at high energies to a landscape with mainly positive but fluctuating curvature in the collapsed phase. Moreover, curvature fluctuations show a maximum in correspondence with the energy of a possible further transition, occurring at lower energies than the collapse one, whose existence had been previously conjectured on the basis of a local analysis of the energy landscape and whose effect on the usual thermodynamic quantities, if any, is extremely weak. We also estimate the largest Lyapunov exponent λ of the SGR using the geometric observables. The geometric estimate always gives the correct order of magnitude of λ and is also quantitatively correct at small energy densities and, in the limit N → ∞, in the whole homogeneous phase.

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“…The λ cl ∝ ǫ 1/3 scaling was already found in the context of motion along a stochastic magnetic field [34], and in the theory of products of random matrices [35,36]. We shall follow these steps as closely as possible for the quantum case.…”

Section: Classical Casementioning

confidence: 71%

Out-of-time-order correlator in weakly perturbed integrable systems

Goldfriend,

Kurchan

2019

Preprint

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Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time, but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent -defined by the evolution of the 4-point out-of-time-order correlator (OTOC) -of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. We find that i) in the semi-classical limit the quantum Lyapunov exponent is given by the classical one: it scales as ǫ 1/3 , with ǫ being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is ∼ ǫ −1 ). ii) in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and iii) for sufficiently small perturbations the ǫ 1/3 dependence is also suppressed -another purely quantum effect which we explain. Several numerical examples which demonstrate the theoretical predictions are given. The implication for the results to the behavior of real near-integrable systems, and for quantum limits on chaos are briefly discussed.

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Generalized Lyapunov exponents of the random harmonic oscillator: Cumulant expansion approach

Cited by 10 publications

References 64 publications

Geometry of the energy landscape of the self-gravitating ring

Geometry of the energy landscape of the self-gravitating ring

Geometry of the energy landscape of the self-gravitating ring

Out-of-time-order correlator in weakly perturbed integrable systems

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