Statistics of finite-time Lyapunov exponents in a random time-dependent potential

Schomerus, Henning; Titov, M.

doi:10.1103/physreve.66.066207

Cited by 65 publications

(98 citation statements)

References 47 publications

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“…This equation appears frequently in the theory of one-dimensional localization, and in the related problem of the harmonic oscillator with randomly diffusing frequency (see References [9,4,24,13,17], whose approches we shall follow).…”

Section: A Single Degree Of Freedommentioning

confidence: 99%

Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems

Lam

Kurchan

2014

J Stat Phys

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Integrable non-linear Hamiltonian systems perturbed by additive noise develop a Lyapunov instability, and are hence chaotic, for any amplitude of the perturbation. This phenomenon is related, but distinct, from Taylor's diffusion in hydrodynamics. We develop expressions for the Lyapunov exponents for the cases of white and colored noise. The situation described here being 'multi-resonance' -by nature well beyond the Kolmogorov-Arnold-Moser regime, it offers an analytic glimpse on the regime in which many near-integrable systems, such as some planetary systems, find themselves in practice. We show with the aid of a simple example, how one may model in some cases weakly chaotic deterministic systems by a stochastically perturbed one, with good qualitative results.

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Section: A Single Degree Of Freedommentioning

confidence: 99%

Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems

Lam

Kurchan

2014

J Stat Phys

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“…Since fluctuations of t 1 \η\δχ(ί)/δχ(ΰ)\ decrease hke t 1/2 , the Lyapunov exponent X 0 is self-aveiagmg [7], while the \ ; 's aie not Foi an analytical descnption, we stau fiom the Gaussian one-dimensional wave packet 1/4 = |:ZFJ ex P 2h (5) The wave packet is centeied at the pomt x 0 (t),p 0 (t) which moves along a classical tiajectoiy Imtially, ß(t = 0) = 0 and α(ί = 0) = l Diveigence of trajectones leads to the exponen tial bioadening of the packet, thus a(f)«exp(-2λί) Since a<äl foi t>l/\, the wave packet in phase space becomes highly elongated with length /||= \Μ·(1 + ß 2~) /a and width / ± = Ä//|| The paiametei β = Δρ/Δχ icpiesents the tilt angle of the elongated wave packet [8] The Gaussian appioximation (5) bieaks down at the Ehienfest time r £ =|-X~1|lnA| when /u becomes of the oidei of the size of the System…”

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confidence: 99%

Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics

2003

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We leexamine the pioblem of the ' Loschmidt echo " that measures the sensitivily lo perturbaüon of quantum-chaoüc dynamics The overlap squaied M(t) of two wave packets evolving under shghtly diffeient Hamiltoman is shown to have the double-exponential initial decay ^exp(-constantXe 2Xo ') in the mam part of the phase space The coefficient X 0 is the seif averagmg Lyapunov exponent The aveiage decay M*e λ '' is single exponential with a different coefficient λ ] The volume of phase space that contnbutes to M vanishes in the classical hmit Ä-»0 for times less than the Ehienfest time T E = jX^'|ln h\ It is only after the Ehrenfest time that the average decay is representative for a typical initial condition is not constiamed by umtanty Jalabert and Pastawski [4] discoveied that M(t) (which they referred to äs the "Loschmidt echo") decays ccg~x' if ψ 0 is a nanow wave packet in a chaotic legion of phase space, pioviding an appealmg connection between classical and quantum chaosThe discoveiy of Jalabeit and Pastawski gave a new im petus [5] to what Schack and Caves called "hypei sensitivity to peituibations" of quantum-chaotic dynamics The piesent papei diffeis fiom this body of hteiatuie in that we consider the statisttcs of M(t) äs ψ 0 vanes ovei the chaotic phase space We find that the aveiage decay M(t)<^e~^' is due to legions of phase space that become vamshmgly small m the classical hmit Ä efl ->0 (The effective Planck constant A eff = h/S 0 is set by the inveise of a typical action S 0 ) The dominant decay is a double exponential ocexp(-constant Xe 2X '), so it is tiuly hypei sensitive The slowet singleexponential decay is lecoveied at the Ehienfest time T E Befoie piesentmg oui analytical theoiy, we show in Fig l the data fiom a numencal Simulation that illustiates the hypersensitivity mentioned above The Hamiltoman is the quantum kicked lotatoi [1]The peituibed Hamiltoman H' = H+SH is obtamed by the leplacement Κ-^Κ+δΚ The coordmate χ is penodic, χ =χ + 2ττ Το woik with a finite dimensional Hilbert space, we discretize

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“…(34). Firstly, since ∆p 0 (1) decreases with increasing σ, when σ is large enough, the right hand side of Eq.…”

Section: Time Interval T < τ1mentioning

confidence: 99%

“…Thirdly, in the Lyapunov regime, the decay rate of average fidelity has been found different from the Lyapunov exponent, although still perturbation-independent, in systems possessing large fluctuation in the finite-time Lyapunov exponent [33,34], as in the kicked top and kicked rotator models [9,19,20]. A semiclassical WKB description of wave packets suggests an exp(−λ 1 t) decay for the fidelity, with λ 1 < λ [20].…”

Section: Introductionmentioning

confidence: 99%

Uniform semiclassical approach to fidelity decay: From weak to strong perturbation

Wang

2005

Phys. Rev. E

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We study fidelity decay by a uniform semiclassical approach, in the three perturbation regimes, namely, the perturbative regime, the Fermi-golden-rule (FGR) regime, and the Lyapunov regime. A semiclassical expression is derived for fidelity of initial Gaussian wave packets with width of the order √h (h being the effective Planck constant). Short time decay of fidelity of initial Gaussian wave packets is also studied, with respect to two time scales introduced in the semiclassical approach. In the perturbative regime, it is confirmed numerically that fidelity has the FGR decay before the Gaussian decay sets in. An explanation is suggested to a non-FGR decay in the FGR regime, which has been observed in a system with weak chaos in the classical limit, by using the Levy distribution as an approximation for the distribution of action difference. In the Lyapunov regime, it is shown that the average of the logarithm of fidelity may have roughly the Lyapunov decay within some time interval, in systems possessing large fluctuation of the finite-time Lyapunov exponent in the classical limit.

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Statistics of finite-time Lyapunov exponents in a random time-dependent potential

Cited by 65 publications

References 47 publications

Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems

Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems

Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics

Uniform semiclassical approach to fidelity decay: From weak to strong perturbation

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