Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials

We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from non-analytic potentials. In particular, we study the dynamics of a model governed by a "graphene-type" force law and one inspired by Hollomon's law describing "work-hardening" effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of non-analytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles $N$ and the total energy $E$ are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large $N$, it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.Comment: Accepted for publication in the International Journal of Bifurcation and Chao

Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basin

Transport and mixing properties of surface currents can be detected from altimetric data by both Eulerian and Lagrangian diagnostics. In contrast with Eulerian diagnostics, Lagrangian tools like the local Lyapunov exponents have the advantage of exploiting both spatial and temporal variability of the velocity field and are in principle able to unveil subgrid filaments generated by chaotic stirring. However, one may wonder whether this theoretical advantage is of practical interest in real-data, mesoscale and submesoscale analysis, because of the uncertainties and resolution of altimetric products, and the non-passive nature of biogeochemical tracers. Here we compare the ability of standard Eulerian diagnostics and the finite-size Lyapunov exponent in detecting instantaneaous and climatological transport and mixing properties. By comparing with sea-surface temperature patterns, we find that the two diagnostics provide similar results for slowly evolving eddies like the first Alboran gyre. However, the Lyapunov exponent is also able to predict the (sub-)mesoscale filamentary process occuring along the Algerian current and above the Balearic Abyssal Plain. Such filaments are also observed, with some mismatch, in sea-surface temperature patterns. Climatologies of Lyapunov exponents do not show any compact relation with other Eulerian diagnostics, unveiling a different structure even at the basin scale. We conclude that filamentation dynamics can be detected by reprocessing available altimetric data with Lagrangian tools, giving insight into (sub-)mesoscale stirring processes relevant to tracer observations and complementing traditional Eulerian diagnostics

Billiards correlation functions

We discuss various experiments on the time decay of velocity autocorrelation functions in billiards. We perform new experiments and find results which are compatible with an exponential mixing hypothesis, first put forward by [FM]: they do not seem compatible with the stretched exponentials believed, in spite of [FM], to describe the mixing. The analysis led us to several byproducts: we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in time $t_m$ (as t_m\to\io) finding a strong dependence on some geometric characteristics of the locus of the billiards obstacles.Comment: 25 pages, 27 figures, POSTSCRIPT, not encoded, 730K. Keywords: Billiards, correlation functions, velocity autocorrelation, diffusion coefficients, Lorentz model, mixing, ergodic theory, chaos, Lyapunov exponents, numerical experiment

Lagrangian statistics of particle pairs in homogeneous isotropic turbulence

We present a detailed investigation of the particle pair separation process in homogeneous isotropic turbulence. We use data from direct numerical simulations up to Taylor's Reynolds number 280 following the evolution of about two million passive tracers advected by the flow over a time span of about three decades. We present data for both the separation distance and the relative velocity statistics. Statistics are measured along the particle pair trajectories both as a function of time and as a function of their separation, i.e. at fixed scales. We compare and contrast both sets of statistics in order to gain an insight into the mechanisms governing the separation process. We find very high levels of intermittency in the early stages, that is, for travel times up to order ten Kolmogorov time scales. The fixed scale statistics allow us to quantify anomalous corrections to Richardson diffusion in the inertial range of scales for those pairs that separate rapidly. It also allows a quantitative analysis of intermittency corrections for the relative velocity statistics.Comment: 16 pages, 16 figure

The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current

This article reviews several recently developed Lagrangian tools and shows how their combined use succeeds in obtaining a detailed description of purely advective transport events in general aperiodic flows. In particular, because of the climate impact of ocean transport processes, we illustrate a 2D application on altimeter data sets over the area of the Kuroshio Current, although the proposed techniques are general and applicable to arbitrary time dependent aperiodic flows. The first challenge for describing transport in aperiodical time dependent flows is obtaining a representation of the phase portrait where the most relevant dynamical features may be identified. This representation is accomplished by using global Lagrangian descriptors that when applied for instance to the altimeter data sets retrieve over the ocean surface a phase portrait where the geometry of interconnected dynamical systems is visible. The phase portrait picture is essential because it evinces which transport routes are acting on the whole flow. Once these routes are roughly recognised it is possible to complete a detailed description by the direct computation of the finite time stable and unstable manifolds of special hyperbolic trajectories that act as organising centres of the flow.Comment: 40 pages, 24 figure

An experimental test of the local fluctuation theorem in chains of weakly interacting Anosov systems

An experimental test of a large fluctuation theorem is performed on a chain of coupled ``cat maps''. Our interest is focused on the behavior of a subsystem of this chain. A local entropy creation rate is defined and we show that the local fluctuation theorem derived in [G1] is experimentally observable already for small subsystems.Comment: LaTeX, 15 pages, 5 figure

Hamiltonian dynamics and geometry of phase transitions in classical XY models

The Hamiltonian dynamics associated to classical, planar, Heisenberg XY models is investigated for two- and three-dimensional lattices. Besides the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively new information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests to consider other observables of geometric meaning tightly related with the largest Lyapunov exponent. The numerical computation of these observables - unusual in the study of phase transitions - sheds a new light on the microscopic dynamical counterpart of thermodynamics also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces $\Sigma_E$ of phase space can be naturally established. In this framework, an approximate formula is worked out, determining a highly non-trivial relationship between temperature and topology of the $\Sigma_E$. Whence it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of the $\Sigma_E$. This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.Comment: in press on Physical Review E, 43 pages, LaTeX (uses revtex), 22 PostScript figure

Lagrangian and inertial transport in atmospheric and chaotic flows

This thesis presents a compendium of publications related to transport studies analyzed from the perspective of dynamical systems. The goal is to address the role that particle properties and the flow have on the organization of trajectories and hence the transport. To observe how transport is structured, we focus on the most widely used method: the Finite Time Lyapunov Exponents. These exponents measure the separation rate of the particles starting from nearby initial positions, estimating the hyperbolicity of the trajectories. This method allows us to make a first approach to the problem, obtaining the borders or frontiers between regions with different dynamics given a simplified vision of transport. The transport structures related with this method, are called Lagrangian Coherent Structures. In the first study, the Lagrangian transport in the troposphere was analyzed. The atmospheric flow is characterized by being turbulent in a continuum of spatiotemporal scales. Within these scales, it was observed that there are structures such as the Atmospheric Rivers that maintain a spatial and temporal coherence of the order of days acting as organizers of water vapor transport and therefore dominating the dynamics of the region at the moment they occur. At the same time, the persistence and repetition of these structures, together with all the other tropospheric structures, introduce mixing into the atmosphere. Those areas in middle latitudes where these structures develop have higher mixing variability. This is mainly due to seasonal changes. However, those regions with less variability, such as the equatorial zones, the mixing and its variability on day scales, are mainly associated with inter-annual variability events such as El Ni ˜no or La Ni ˜na or the Intertropical Convergence Zone (ITCZ). In addition, the mixing information of the air masses from a climatic point of view, was used as a predictor of rainfall for the Iberian region. The Atlantic margin is characterized by an intense activity of Atmospheric Rivers, being one of the main causes of precipitation. However, the problem of determining the activity of rainfall months in advance is complex, for this reason the use of new variables as potential predictors is required. It has been obtained that the mixing, in the Atlantic region, is related to the precipitation on the Iberian Peninsula. Addressing on the second study, we focus on the influence of forces on the particles motion so the resolution of motion equation is required to obtain the trajectories they describe. The particles are modeled as small spheres with mass, but the fact that their movement is decoupled from the flow makes their trajectories depend initially on other properties such as the initial velocity. It was observed that this dependence, for certain flows, is even higher than small perturbations in its position, mainly in those regions where there is a high spatial variability of the fluid such as regions with shear. The same happens for bubbles where flotation effects appear. They are very sensitive to the inertial effects and especially to the disturbances of the radius as well as the effects of merging with other bubbles, being especially relevant in the initial instants of the movement. In addition, it has been observed that particles properties and their collective motion play a key role in the synchronization of finite-size chemical oscillators. To experimentally support some of the aforementioned behaviors, experimental data are needed to measure the trajectories of the particles. Particle Tracking Velocimetry (PTV) methods, track the trajectories of individual particles in three-dimensional space. In the last part of this thesis, we present an experimental setup and some preliminary results of trajectories of the particles mentioned above in a high turbulent flow