Spectral characterization of anomalous diffusion of a periodic piecewise linear intermittent map

Abstract

For a piecewise linear version of the periodic map with anomalous diffusion, the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the Frobenius-Perron operator are explicitly derived. The evolution of the averages is controlled by real eigenvalues as well as continuous spectra terminating at the unit circle. Appropriate scaling limits are shown to give a normal diffusion if the reduced map is in the stationary regime with normal fluctuations, a Lévy flight if the reduced map is in the stationary regime with Lévy-type fluctuations and a transport of ballistic type if the reduced map is in the non-stationary regime.

Spectral analysis and an area-preserving extension of a piecewise linear intermittent map

“…And we define a set X O as the functional space which consists of such observables. This functional space is invariant under the action of the adjoint of the FP operatorP * , namely,P Furthermore we restrict initial densities to be piecewise constant [24],…”

“…Note that this space is dense in the Hilbert space L 2 [0, 1] of the square integrable functions on [0, 1]. Furthermore we restrict initial densities to be piecewise constant [24],…”

“…Therefore {F in , F d , F λ } are eigenfunctions of the FP operator in a generalized sense [19,24,29,30]. On the other hand, the right eigenfunctions { Fin , Fd , Fλ } satisfy the relations ( P * Fin , ρ) ≡ ( Fin , P ρ) = ( Fin , ρ),…”

“…For understanding sub-exponential decay of correlation functions in dynamical systems, non-hyperbolic 1-dimensional maps have been studied by several authors [17,18,19,20,21,22,23,24,25,26] and they have found power law decay of correlations in their models. Therefore, it is natural to imagine connections of these non-hyperbolic maps and mixed type Hamiltonian systems, however, extensions of these maps to 2-dimensional area-preserving systems are unknown.…”

Transport Properties of a Piecewise Linear Transformation and Deterministic Levy Flights

“…Deterministic models of anomalous diffusion have been introduced and investigated by several researchers. 12), 17) In those models, the flight lengths are finite, there are no explicit extensions to area preserving systems, and the origin of anomalous transport phenomena is the existence of long time correlations. Contrastingly, the model discussed in this paper generates arbitrarily long flights, which are essential for Lévy flight.…”