Marie-Christine Firpo

Phase transition in the collisionless damping regime for wave-particle interaction

Gibbs statistical mechanics is derived for the Hamiltonian system coupling a wave to N particles self-consistently. This identifies Landau damping with a regime where a second order phase transition occurs. For nonequilibrium initial data with warm particles, a critical initial wave intensity is found: above it, thermodynamics predicts a finite wave amplitude in the limit N-->infinity; below it, the equilibrium amplitude vanishes. Simulations support these predictions providing new insight into the long-time nonlinear fate of the wave due to Landau damping in plasmas.

Long-time discrete particle effects versus kinetic theory in the self-consistent single-wave model.

The influence of the finite number N of particles coupled to a monochromatic wave in a collisionless plasma is investigated. For growth as well as damping of the wave, discrete particle numerical simulations show an N-dependent long time behavior resulting from the dynamics of individual particles. This behavior differs from the one due to the numerical errors incurred by Vlasov approaches. Trapping oscillations are crucial to long time dynamics, as the wave oscillations are controlled by the particle distribution inhomogeneities and the pulsating separatrix crossings drive the relaxation towards thermal equilibrium.

Kinetic limit of N-body description of wave-particle self-consistent interaction

AbstractA system of N particles n

Chaos suppression in the large size limit for long-range systems

xi ^N = x_1 ,upsilon_1,...,x_N ,upsilon _N )

Trapping oscillations, discrete particle effects and kinetic theory of collisionless plasma

n interacting self-consistently with one wave Z = A exp(iφ) is considered. Given initial data (Z(N)(0), ξN(0)), it evolves according to Hamiltonian dynamics to (Z(N)(t), ξN(t)). In the limit N → ∞, this generates a Vlasov-like kinetic equation for the distribution function f(x, v, t), abbreviated as f(t), coupled to the envelope equation for Z: initial data (Z(∞)(0), f(0)) evolve to (Z(∞)(t), f(t)). The solution (Z, f) exists and is unique for any initial data with finite energy. Moreover, for any time T>0, given a sequence of initial data with N particles distributed so that the particle distribution fN(0) → f(0) weakly and with Z(N)(0) → Z(0) as N → ∞, the states generated by the Hamiltonian dynamics at all times 0 ≤ t ≤ T are such that (Z(N)(t), fN(t)) converges weakly to (Z(∞)(t), f(t)).

Finite-N Effects, Phase Transition, and Long-Time Evolution of the Wave-Particle System

A study of coherent structures and self-consistent transport is presented in the context of a Hamiltonian mean field, single wave model. The model describes the weakly nonlinear dynamics of marginally stable plasmas and fluids, and it is related to models of systems with long-range interactions in statistical mechanics. In plasma physics the model applies to the interaction of electron holes and electron clumps, which are depletions and excesses of phase-space electron density with respect to a fixed background. In fluid dynamics the system describes the interaction of vortices with positive and negative circulation in a two-dimensional background shear flow. Numerical simulations in the finite-N and in the N--> infinity kinetic limit (where N is the number of particles) show the existence of coherent, rotating dipole states. We approximate the dipole as two macroparticles (one hole and one clump) and consider the N=2 limit of the model. We show that this limit has a family of symmetric, rotating integrable solutions described by a one-degree-of-freedom nontwist Hamiltonian. A perturbative solution of the nontwist Hamiltonian provides an accurate description of the mean field and rotation period of the dipole. The coherence of the dipole is explained in terms of a parametric resonance between the rotation frequency of the macroparticles and the oscillation frequency of the self-consistent mean field. This resonance creates islands of integrability that shield the dipole from regions of chaotic transport. For a class of initial conditions, the mean field exhibits an elliptic-hyperbolic bifurcation that leads to the filamentation, chaotic mixing and eventual destruction of the dipole. (c) 2002 American Institute of Physics.

Kinetic theory and large-N limit for wave-particle self-consistent interaction

We consider the class of long-range Hamiltonian systems first introduced by Anteneodo and Tsallis and called the α-XY model. This involves N classical rotators on a d-dimensional periodic lattice interacting all to all with an attractive coupling whose strength decays as r -α, r being the distance between sites. Using a recent geometrical approach, we estimate for any d-dimensional lattice the scaling of the largest Lyapunov exponent (LLE) with N as a function of α in the large energy regime where rotators behave almost freely. We find that the LLE vanishes as N -κ, with κ = 1/3 for 0≤α/d≤1/2 and κ = 2/3(1-α/d) for 1/2≤α/d<1. These analytical results present a nice agreement with numerical results obtained by Campa et al, including deviations at small N.

O'Neil's threshold for nonlinear Landau damping and phase transition in the wave-particle system

A dynamical analysis is presented that self-consistently takes into account the motion of the critical layer, in which the magnetic field reconnects, to describe how the m=n=1 resistive internal kink mode develops in the nonlinear regime. The amplitude threshold marking the onset of strong nonlinearities due to a balance between convective and mode coupling terms is identified. We predict quantitatively the early nonlinear growth rate of the m=n=1 mode below this threshold.