1,195 research outputs found

### Classical dynamics on graphs

We consider the classical evolution of a particle on a graph by using a
time-continuous Frobenius-Perron operator which generalizes previous
propositions. In this way, the relaxation rates as well as the chaotic
properties can be defined for the time-continuous classical dynamics on graphs.
These properties are given as the zeros of some periodic-orbit zeta functions.
We consider in detail the case of infinite periodic graphs where the particle
undergoes a diffusion process. The infinite spatial extension is taken into
account by Fourier transforms which decompose the observables and probability
densities into sectors corresponding to different values of the wave number.
The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a
Frobenius-Perron operator corresponding to a given sector. The diffusion
coefficient is obtained from the hydrodynamic modes of diffusion and has the
Green-Kubo form. Moreover, we study finite but large open graphs which converge
to the infinite periodic graph when their size goes to infinity. The lifetime
of the particle on the open graph is shown to correspond to the lifetime of a
system which undergoes a diffusion process before it escapes.Comment: 42 pages and 8 figure

### Chaotic and fractal properties of deterministic diffusion-reaction processes

We study the consequences of deterministic chaos for diffusion-controlled
reaction. As an example, we analyze a diffusive-reactive deterministic
multibaker and a parameter-dependent variation of it. We construct the
diffusive and the reactive modes of the models as eigenstates of the
Frobenius-Perron operator. The associated eigenvalues provide the dispersion
relations of diffusion and reaction and, hence, they determine the reaction
rate. For the simplest model we show explicitly that the reaction rate behaves
as phenomenologically expected for one-dimensional diffusion-controlled
reaction. Under parametric variation, we find that both the diffusion
coefficient and the reaction rate have fractal-like dependences on the system
parameter.Comment: 14 pages (revtex), 12 figures (postscript), to appear in CHAO

### Quantum Hall-like effect for cold atoms in non-Abelian gauge potentials

We study the transport of cold fermionic atoms trapped in optical lattices in
the presence of artificial Abelian or non-Abelian gauge potentials. Such
external potentials can be created in optical lattices in which atom tunneling
is laser assisted and described by commutative or non-commutative tunneling
operators. We show that the Hall-like transverse conductivity of such systems
is quantized by relating the transverse conductivity to topological invariants
known as Chern numbers. We show that this quantization is robust in non-Abelian
potentials. The different integer values of this conductivity are explicitly
computed for a specific non-Abelian system which leads to a fractal phase
diagram.Comment: 6 pages, 2 figure

### Viscosity in the escape-rate formalism

We apply the escape-rate formalism to compute the shear viscosity in terms of
the chaotic properties of the underlying microscopic dynamics. A first passage
problem is set up for the escape of the Helfand moment associated with
viscosity out of an interval delimited by absorbing boundaries. At the
microscopic level of description, the absorbing boundaries generate a fractal
repeller. The fractal dimensions of this repeller are directly related to the
shear viscosity and the Lyapunov exponent, which allows us to compute its
values. We apply this method to the Bunimovich-Spohn minimal model of viscosity
which is composed of two hard disks in elastic collision on a torus. These
values are in excellent agreement with the values obtained by other methods
such as the Green-Kubo and Einstein-Helfand formulas.Comment: 16 pages, 16 figures (accepted in Phys. Rev. E; October 2003

### On the long-time behavior of spin echo and its relation to free induction decay

It is predicted that (i) spin echoes have two kinds of generic long-time
decays: either simple exponential, or a superposition of a monotonic and an
oscillatory exponential decays; and (ii) the long-time behavior of spin echo
and the long-time behavior of the corresponding homogeneous free induction
decay are characterized by the same time constants. This prediction extends to
various echo problems both within and beyond nuclear magnetic resonance.
Experimental confirmation of this prediction would also support the notion of
the eigenvalues of time evolution operators in large quantum systems.Comment: 4 pages, 4 figure

### Quantum fingerprints of classical Ruelle-Pollicot resonances

N-disk microwave billiards, which are representative of open quantum systems,
are studied experimentally. The transmission spectrum yields the quantum
resonances which are consistent with semiclassical calculations. The spectral
autocorrelation of the quantum spectrum is shown to be determined by the
classical Ruelle-Pollicot resonances, arising from the complex eigenvalues of
the Perron-Frobenius operator. This work establishes a fundamental connection
between quantum and classical correlations in open systems.Comment: 6 pages, 2 eps figures included, submitted to PR

### Complexity and non-separability of classical Liouvillian dynamics

We propose a simple complexity indicator of classical Liouvillian dynamics,
namely the separability entropy, which determines the logarithm of an effective
number of terms in a Schmidt decomposition of phase space density with respect
to an arbitrary fixed product basis. We show that linear growth of separability
entropy provides stricter criterion of complexity than Kolmogorov-Sinai
entropy, namely it requires that dynamics is exponentially unstable, non-linear
and non-markovian.Comment: Revised version, 5 pages (RevTeX), with 6 pdf-figure

### Scarring in open quantum systems

We study scarring phenomena in open quantum systems. We show numerical
evidence that individual resonance eigenstates of an open quantum system
present localization around unstable short periodic orbits in a similar way as
their closed counterparts. The structure of eigenfunctions around these
classical objects is not destroyed by the opening. This is exposed in a
paradigmatic system of quantum chaos, the cat map.Comment: 4 pages, 4 figure

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