Coexisting crystal and liquid-like properties in a 2D long-range self-consistent model.

A two-dimensional class of mean-field models serving as a minimal frame to study long-range interaction in two space dimensions is considered. In the case of an anisotropic mixed attractive-repulsive interaction, an initially spatially homogeneous cold fluid is dynamically unstable and evolves towards a quasi-stationary state in which the less energetic particles get trapped into clusters forming a Bravais-like lattice, mimicking a crystalline state. Superimposed to this, one observes in symplectic numerical simulations a flux of slightly more energetic particles channeling through this crystalline background. The resultant system combines the rigidity features of a solid, as particles from a displaced core are shown to snap back into place after a transient, and the dynamical diffusive features of a liquid for the fraction of channeling and free particles. The combination of solid and liquid properties is numerically observed here within the classical context. The quantum transposition of the model may be experimentally reached using the latest ultracold atoms techniques to generate long-range interactions.

Stochastic treatment of finite-N effects in mean-field systems and its application to the lifetimes of coherent structures.

A stochastic treatment yielding to the derivation of a general Fokker-Planck equation is presented to model the slow convergence toward equilibrium of mean-field systems due to finite-N effects. The thermalization process involves notably the disintegration of coherent structures that may sustain out-of-equilibrium quasistationary states. The time evolution of the fraction of particles remaining close to a mean-field potential trough is analytically computed. This indicator enables to estimate the lifetime of coherent structures and thermalization time scale in mean-field systems.

Early out-of-equilibrium beam-plasma evolution.

We solve analytically the out-of-equilibrium initial stage that follows the injection of a radially finite electron beam into a plasma at rest and test it against particle-in-cell simulations. For initial large beam edge gradients and not too large beam radius, compared to the electron skin depth, the electron beam is shown to evolve into a ring structure. For low enough transverse temperatures, the filamentation instability eventually proceeds and saturates when transverse isotropy is reached. The analysis accounts for the variety of very recent experimental beam transverse observations.

Interplay of collisions with quasilinear growth rates of relativistic electron-beam-driven instabilities in a superdense plasma.

We focus attention on the rapidly growing electromagnetic instabilities arising in the interaction of intense and relativistic electron beams (REB) with supercompressed thermonuclear fuel. REB-target system is considered neutralized in charge and current with a distribution function including beam and target temperatures. The electromagnetic filamentation (Weibel) instability is first considered analytically in a linear approximation. Relevant growth rates parameters then highlight density ratios between target and particle beams, as well as transverse temperatures. Significant refinements include mode-mode coupling and collisions with target electrons. The former qualify the so-called quasilinear (weakly turbulent) approach. Usually, it produces significantly lower growth rates than the linear ones. Collisions enhance them slightly for kc/omega(p) < 1, and dampen them strongly for kc/omega(p) < 1. In a low temperature target plasma, intrabeam scattering also contributes to the instability taming, while keeping it close to zero in a warm plasma. Our numerical exploration provides further support to the cone-angle configuration (Osaka experiment) with REB penetrating close to the dense core of superdense deuterium + tritium fuel.

Electromagnetic instabilities for relativistic beam-plasma interaction in whole k space: nonrelativistic beam and plasma temperature effects.

For the system formed by a relativistic electron beam and its plasma return current, we investigate the effects of both transverse and parallel beam and plasma temperatures on the linear stability of collective electromagnetic modes. We focus on nonrelativistic temperatures and wave-vector orientations ranging from two-stream to filamentation instabilities. Water-bag distributions are used to model temperature effects and we discuss their relevance. Labeling Theta(k) the angle between the beam and the wave vector, one or two critical angles Theta(c,i) are determined exactly and separate the k space into two parts. Modes with Theta(k) < Theta(c) =min ( Theta(c,i)) are quasilongitudinal and poorly affected by any kind of temperature. Modes having Theta(k) > Theta(c) are very sensitive to transverse beam and plasma parallel temperatures. Also, parallel plasma temperature can trigger a transition between the beam-dependent filamentation instability (Theta(k) =pi/2) and the plasma-temperature-dependent Weibel instability so that two-stream, filamentation, and Weibel instabilities are eventually closely connected to each other. The maximum growth rate being reached for a mode with Theta(k) < Theta(c), no temperature of any kind can significantly reduce it in the nonrelativistic temperature regime.

Characterization of the initial filamentation of a relativistic electron beam passing through a plasma.

The linear instability that induces a relativistic electron beam passing through a plasma with return current to filament transversely is often related to some filamentation mode with the wave vector normal to the beam or confused with Weibel modes. We show that these modes may not be relevant in this matter and identify the most unstable mode on the two-stream or filamentation branch as the main trigger for filamentation. This sets both the characteristic transverse and longitudinal filamentation scales in the nonresistive initial stage.

Collective electromagnetic modes for beam-plasma interaction in the whole k space.

We investigate the linear stability of the system formed by an electron beam and its return plasma current within a general framework, namely, for any orientation of the wave vector k with respect to the beam and without any a priori assumption on the orientation of the electric field with respect to k . We apply this formalism to three configurations: cold beam and cold plasma, cold beam and hot plasma, and cold relativistic beam and hot plasma. We proceed to the identification and systematic study of the two branches of the electromagnetic dispersion relation. One pertains to Weibel-like beam modes with transverse electric proper waves. The other one refers to electric proper waves belonging to the plane formed by k and the beam, it divides between Weibel-like beam modes and a branch sweeping from longitudinal two-stream modes to purely transverse filamentation modes. For this latter branch, we thoroughly investigate the intermediate regime between two-stream and filamentation instabilities for arbitrary wave vectors. When some plasma temperature is allowed for, the system exhibits a critical angle at which waves are unstable for every k . Besides, in the relativistic regime, the most unstable mode on this branch is reached for an oblique wave vector. This study is especially relevant to the fast ignition scenario as its generality could help clarify some confusing linear issues of present concern. This is a prerequisite towards more sophisticated nonlinear treatments.

Dynamical analysis of the nonlinear growth of the m = n = 1 resistive internal mode.

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.

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.