974 research outputs found
A trivial observation on time reversal in random matrix theory
It is commonly thought that a state-dependent quantity, after being averaged
over a classical ensemble of random Hamiltonians, will always become
independent of the state. We point out that this is in general incorrect: if
the ensemble of Hamiltonians is time reversal invariant, and the quantity
involves the state in higher than bilinear order, then we show that the
quantity is only a constant over the orbits of the invariance group on the
Hilbert space. Examples include fidelity and decoherence in appropriate models.Comment: 7 pages 3 figure
Phase transitions as topology changes in configuration space: an exact result
The phase transition in the mean-field XY model is shown analytically to be
related to a topological change in its configuration space. Such a topology
change is completely described by means of Morse theory allowing a computation
of the Euler characteristic--of suitable submanifolds of configuration
space--which shows a sharp discontinuity at the phase transition point, also at
finite N. The present analytic result provides, with previous work, a new key
to a possible connection of topological changes in configuration space as the
origin of phase transitions in a variety of systems.Comment: REVTeX file, 5 pages, 1 PostScript figur
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
This paper deals with the problem of analytically computing the largest
Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is
succesfully reached within a theoretical framework that makes use of a
geometrization of newtonian dynamics in the language of Riemannian geometry. A
new point of view about the origin of chaos in these systems is obtained
independently of homoclinic intersections. Chaos is here related to curvature
fluctuations of the manifolds whose geodesics are natural motions and is
described by means of Jacobi equation for geodesic spread. Under general
conditions ane effective stability equation is derived; an analytic formula for
the growth-rate of its solutions is worked out and applied to the
Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent
agreement is found the theoretical prediction and the values of the Lyapunov
exponent obtained by numerical simulations for both models.Comment: RevTex, 40 pages, 8 PostScript figures, to be published in Phys. Rev.
E (scheduled for November 1996
Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics
We re-examine the problem of the "Loschmidt echo", which measures the
sensitivity to perturbation of quantum chaotic dynamics. The overlap squared
of two wave packets evolving under slightly different Hamiltonians is
shown to have the double-exponential initial decay in the main part of phase space. The
coefficient is the self-averaging Lyapunov exponent. The average
decay is single exponential with a different
coefficient . The volume of phase space that contributes to
vanishes in the classical limit for times less than the
Ehrenfest time . It is only after
the Ehrenfest time that the average decay is representative for a typical
initial condition.Comment: 4 pages, 4 figures, [2017: fixed broken postscript figures
Chaos and Complexity of quantum motion
The problem of characterizing complexity of quantum dynamics - in particular
of locally interacting chains of quantum particles - will be reviewed and
discussed from several different perspectives: (i) stability of motion against
external perturbations and decoherence, (ii) efficiency of quantum simulation
in terms of classical computation and entanglement production in operator
spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing,
and (iv) computation of quantum dynamical entropies. Discussions of all these
criteria will be confronted with the established criteria of integrability or
quantum chaos, and sometimes quite surprising conclusions are found. Some
conjectures and interesting open problems in ergodic theory of the quantum many
problem are suggested.Comment: 45 pages, 22 figures, final version, at press in J. Phys. A, special
issue on Quantum Informatio
Evolution of entanglement under echo dynamics
Echo dynamics and fidelity are often used to discuss stability in quantum
information processing and quantum chaos. Yet fidelity yields no information
about entanglement, the characteristic property of quantum mechanics. We study
the evolution of entanglement in echo dynamics. We find qualitatively different
behavior between integrable and chaotic systems on one hand and between random
and coherent initial states for integrable systems on the other. For the latter
the evolution of entanglement is given by a classical time scale. Analytic
results are illustrated numerically in a Jaynes Cummings model.Comment: 5 RevTeX pages, 3 EPS figures (one color) ; v2: considerable revision
;inequality proof omitte
Geometric dynamical observables in rare gas crystals
We present a detailed description of how a differential geometric approach to
Hamiltonian dynamics can be used for determining the existence of a crossover
between different dynamical regimes in a realistic system, a model of a rare
gas solid. Such a geometric approach allows to locate the energy threshold
between weakly and strongly chaotic regimes, and to estimate the largest
Lyapunov exponent. We show how standard mehods of classical statistical
mechanics, i.e. Monte Carlo simulations, can be used for our computational
purposes. Finally we consider a Lennard Jones crystal modeling solid Xenon. The
value of the energy threshold turns out to be in excellent agreement with the
numerical estimate based on the crossover between slow and fast relaxation to
equilibrium obtained in a previous work by molecular dynamics simulations.Comment: RevTeX, 19 pages, 6 PostScript figures, submitted to Phys. Rev.
A Uniform Approximation for the Fidelity in Chaotic Systems
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.Comment: 16 pages, 4 figures, accepted in J. Phys. A, special edition on
Random Matrix Theor
Active Galactic Nuclei under the scrutiny of CTA
Active Galactic Nuclei (hereafter AGN) produce powerful outflows which offer
excellent conditions for efficient particle acceleration in internal and
external shocks, turbulence, and magnetic reconnection events. The jets as well
as particle accelerating regions close to the supermassive black holes
(hereafter SMBH) at the intersection of plasma inflows and outflows, can
produce readily detectable very high energy gamma-ray emission. As of now, more
than 45 AGN including 41 blazars and 4 radiogalaxies have been detected by the
present ground-based gamma-ray telescopes, which represents more than one third
of the cosmic sources detected so far in the VHE gamma-ray regime. The future
Cherenkov Telescope Array (CTA) should boost the sample of AGN detected in the
VHE range by about one order of magnitude, shedding new light on AGN population
studies, and AGN classification and unification schemes. CTA will be a unique
tool to scrutinize the extreme high-energy tail of accelerated particles in
SMBH environments, to revisit the central engines and their associated
relativistic jets, and to study the particle acceleration and emission
mechanisms, particularly exploring the missing link between accretion physics,
SMBH magnetospheres and jet formation. Monitoring of distant AGN will be an
extremely rewarding observing program which will inform us about the inner
workings and evolution of AGN. Furthermore these AGN are bright beacons of
gamma-rays which will allow us to constrain the extragalactic infrared and
optical backgrounds as well as the intergalactic magnetic field, and will
enable tests of quantum gravity and other "exotic" phenomena.Comment: 28 pages, 23 figure
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