Brownian motion and diffusion: from stochastic processes to chaos and beyond

Records and Occupation Time Statistics for Area-Preserving Maps

A relevant problem in dynamics is to characterize how deterministic systems may exhibit features typically associated with stochastic processes. A widely studied example is the study of (normal or anomalous) transport properties for deterministic systems on non-compact phase space. We consider here two examples of area-preserving maps: the Chirikov-Taylor standard map and the Casati-Prosen triangle map, and we investigate transport properties, records statistics, and occupation time statistics. Our results confirm and expand known results for the standard map: when a chaotic sea is present, transport is diffusive, and records statistics and the fraction of occupation time in the positive half-axis reproduce the laws for simple symmetric random walks. In the case of the triangle map, we retrieve the previously observed anomalous transport, and we show that records statistics exhibit similar anomalies. When we investigate occupation time statistics and persistence probabilities, our numerical experiments are compatible with a generalized arcsine law and transient behavior of the dynamics.

Sublinear diffusion in the generalized triangle map

Diffusion of the orbits in a nonchaotic area-preserving map called a generalized triangle map (GTM) is numerically and analytically investigated. We provide accurate empirical evidence that the mean-squared displacement of the momentum for generic perturbation parameter settings increases sublinearly in time, and that the distribution of the momentum obeys a time-fractional diffusion equation. We show that the diffusion properties in the GTM do not follow any of the known stochastic processes generating sublinear diffusion since two seemingly incompatible features, non-Markovian yet stationary, coexist in the dynamics.

Informational analysis of Langevin equation of friction in earthquake rupture processes

In this paper, we analyze the informational properties of time series of slip velocity generated by the Langevin equation of friction in two different frictional regimes: viscous and Coulombian. Representing the generated time series in the Fisher-Shannon plane (where the coordinate axes are the Fisher Information Measure and the Shannon entropy power), the two different frictional regimes are well discriminated. In particular, the viscous regime is characterized by smaller Shannon entropy than the Coulombian one. Furthermore, also the Fisher Information Measure of the slip velocity smoothed by average filter depends on the frictional mechanism, being larger for the viscous regime.

Signatures of chaos in time series generated by many-spin systems at high temperatures

Extracting reliable indicators of chaos from a single experimental time series is a challenging task, in particular, for systems with many degrees of freedom. The techniques available for this purpose often require unachievably long time series. In this paper, we explore a method of discriminating chaotic from multi-periodic integrable motion in many-particle systems. The applicability of this method is supported by our numerical simulations of the dynamics of classical spin lattices at high temperatures. We compared chaotic and nonchaotic regimes of these lattices and investigated the transition between the two. The method is based on analyzing higher-order time derivatives of the time series of a macroscopic observable-the total magnetization of the spin lattice. We exploit the fact that power spectra of the magnetization time series generated by chaotic spin lattices exhibit exponential high-frequency tails, while, for the integrable spin lattices, the power spectra are terminated in a non-exponential way. We have also demonstrated the applicability limits of the above method by investigating the high-frequency tails of the power spectra generated by quantum spin lattices and by the classical Toda lattice.

Subdiffusion due to strange nonchaotic dynamics: a numerical study

We numerically investigate diffusion phenomena in quasiperiodically forced systems with spatially periodic potentials using a lift of the quasiperiodically forced circle map and a quasiperiodically forced damped pendulum. These systems exhibit several types of dynamics: quasiperiodic, strange nonchaotic, and chaotic. The strange nonchaotic and chaotic dynamics induce deterministic diffusion of orbits. The diffusion type gradually changes from logarithmic to subdiffusive within a strange nonchaotic regime and finally becomes normal in a chaotic regime. Fractal time-series analysis shows that the subdiffusion is caused by the antipersistence property of strange nonchaotic motion.

Onset of diffusive behavior in confined transport systems

We investigate the onset of diffusive behavior in polygonal channels for disks of finite size, modeling simple microporous membranes. It is well established that the point-particle case displays anomalous transport, because of slow correlation decay in the absence of defocusing collisions. We investigate which features of point-particle transport survive in the case of finite-sized particles (which undergo defocusing collisions). A similar question was investigated by Lansel, Porter, and Bunimovich [Chaos 16, 013129 (2006)], who found that certain integrals of motion and multiple ergodic components, characteristic of the point-particle case, remain in "mushroom"-like systems with few finite-sized particles. We quantify the time scales over which the transport of disks shows features typical of the point particles, or is driven toward diffusive behavior. In particular, we find that interparticle collisions drive the system toward diffusive behavior more strongly than defocusing boundary collisions. We illustrate how, and at what stage, typical thermodynamic behavior (consistent with kinetic theory) is observed, as particle numbers grow and mean free paths diminish. These results have both applied (e.g., nanotechnological) and theoretical interest.