Xavier Leoncini

Jets, stickiness, and anomalous transport.

Dynamical and statistical properties of the vortex and passive particle advection in chaotic flows generated by 4- and 16-point vortices are investigated. General transport properties of these flows are found to be anomalous and exhibit a superdiffusive behavior with typical second moment exponent mu approximately 1.75. The origin of this anomaly is traced to the presence of coherent structures within the flow, the vortex cores, and the region far from where vortices are located. In the vicinity of these regions stickiness is observed and the motion of tracers is quasiballistic. The chaotic nature of the underlying flow dictates the choice for thorough analysis of transport properties. Passive tracer motion is analyzed by measuring the mutual relative evolution of two nearby tracers. Some tracers travel in each others vicinity for relatively long times. This is related to a hidden order for the tracers, which we call jets. Jets are localized and found in sticky regions. Their structure is analyzed and found to be formed of a nested set of jets within jets. The analysis of the jet trapping time statistics shows a quantitative agreement with the observed transport exponent.

Nonlinear Evolution of q = 1 Triple Tearing Modes in a Tokamak Plasma

In magnetic configurations with two or three q=1 (with q being the safety factor) resonant surfaces in a tokamak plasma, resistive magnetohydrodynamic modes with poloidal mode numbers m much larger than 1 are found to be linearly unstable. It is found that these high-m double or triple tearing modes significantly enhance through nonlinear interactions the growth of the m=1 mode. This may account for the sudden onset of the internal resistive kink, i.e., the fast sawtooth trigger. Based on the subsequent reconnection dynamics that can proceed without formation of the m=1 islands, it is proposed that high-m triple tearing modes are a possible mechanism for precursor-free partial collapses during sawtooth oscillations.

Abundance of Regular Orbits and Nonequilibrium Phase Transitions in the Thermodynamic Limit for Long-Range Systems

We investigate the dynamics of many-body long-range interacting systems, taking the Hamiltonian mean-field model as a case study. We show that regular trajectories, associated with invariant tori of the single-particle dynamics, prevail. The presence of such tori provides a dynamical interpretation of the emergence of long-lasting out-of-equilibrium regimes observed generically in long-range systems. This is alternative to a previous statistical mechanics approach to such phenomena which was based on a maximum entropy principle. Previously detected out-of-equilibrium phase transitions are also reinterpreted within this framework.

Stationary states and fractional dynamics in systems with long range interactions

The dynamics of many-body Hamiltonian systems with long-range interactions is studied, in the context of the so-called α-HMF model. Building on the analogy with the related mean-field model, we construct stationary states of the α-HMF model for which the spatial organization satisfies a fractional equation. At variance, the microscopic dynamics turns out to be regular and explicitly known. As a consequence, dynamical regularity is achieved at the price of strong spatial complexity, namely a microscopic inhomogeneity which locally displays scale invariance.

Out-of-equilibrium solutions in the XY-Hamiltonian Mean-Field model

Out-of-equilibrium magnetised solutions of the XY-Hamiltonian Mean-Field (XY-HMF) model are built using an ensemble of integrable uncoupled pendula. Using these solutions, we display an out-of-equilibrium phase transition using a specific reduced set of the magnetised solutions.

Anomalous transport in Charney-Hasegawa-Mima flows

The transport properties of particles evolving in a system governed by the Charney-Hasegawa-Mima equation are investigated. Transport is found to be anomalous with a nonlinear evolution of the second moments with time. The origin of this anomaly is traced back to the presence of chaotic jets within the flow. All characteristic transport exponents have a similar value around mu = 1.75, which is also the one found for simple point vortex flows in the literature, indicating some kind of universality. Moreover, the law gamma = mu + 1 linking the trapping-time exponent within jets to the transport exponent is confirmed, and an accumulation toward zero of the spectrum of the finite-time Lyapunov exponent is observed. The localization of a jet is performed, and its structure is analyzed. It is clearly shown that despite a regular coarse-grained picture of the jet, the motion within the jet appears as chaotic, but that chaos is bounded on successive small scales.

Existence of quasi-stationary states at the long range threshold

Abstract In this paper the lifetime of quasi-stationary states (QSS) in the α –HMF model are investigated at the long range threshold ( α equals to one). It is found that QSS exist and have a diverging lifetime with system size which scales logaritmically with the number of constituents. This contrast to the exhibited power law below the long range threshold ( α smaller than one) and the observed finite lifetime beyond. Also even beyond this long range threshold the long range nature of the system is displayed, namely the existence of a phase transition. As a consequence of our findings the definition of a long range system is discussed.

HAMILTONIAN DYNAMICS AND THE PHASE TRANSITION OF THE XY MODEL

A Hamiltonian dynamics is defined for the

Chaotic motion of charged particles in toroidal magnetic configurations

\mathrm{XY}

Critical behaviour of the XY -rotors model on regular and small world networks

model by adding a kinetic energy term. Thermodynamical properties (total energy, magnetization, vorticity) derived from microcanonical simulations of this model are found to be in agreement with canonical Monte Carlo results in the explored temperature region. The behavior of the magnetization and energy as functions of the temperature are thoroughly investigated, taking into account finite size effects. By representing the spin field as a superposition of random phased waves, we derive a nonlinear dispersion relation whose solutions allow the computation of thermodynamical quantities, which agree quantitatively with those obtained in numerical experiments, up to temperatures close to the transition. At low temperatures the propagation of phonons is the dominant phenomenon, while above the phase transition the system splits into ordered domains separated by interfaces populated by topological defects. In the high temperature phase, spins rotate, and an analogy with an Ising-like system can be established, leading to a theoretical prediction of the critical temperature