Diffusion limit for many particles in a periodic stochastic acceleration field

The one-dimensional motion of any number \cN of particles in the field of many independent waves (with strong spatial correlation) is formulated as a second-order system of stochastic differential equations, driven by two Wiener processes. In the limit of vanishing particle mass ${\mathfrak{m}} \to 0$, or equivalently of large noise intensity, we show that the momenta of all $N$ particles converge weakly to $N$ independent Brownian motions, and this convergence holds even if the noise is periodic. This justifies the usual application of the diffusion equation to a family of particles in a unique stochastic force field. The proof rests on the ergodic properties of the relative velocity of two particles in the scaling limit.Comment: 20 page

Ornstein-Uhlenbeck limit for the velocity process of an NN-particle system interacting stochastically

An $N$-particle system with stochastic interactions is considered. Interactions are driven by a Brownian noise term and total energy conservation is imposed. The evolution of the system, in velocity space, is a diffusion on a $(3N-1)$-dimensional sphere with radius fixed by the total energy. In the $N\rightarrow\infty$ limit, a finite number of velocity components are shown to evolve independently and according to an Ornstein-Uhlenbeck process.Comment: 19 pages ; streamlined notations ; new section on many particles with momentum conservation ; new appendix on Kac syste

Self-consistency vanishes in the plateau regime of the bump-on-tail instability

Using the Vlasov-wave formalism, it is shown that self-consistency vanishes in the plateau regime of the bump-on-tail instability if the plateau is broad enough. This shows that, in contrast with the "turbulent trapping" Ansatz, a renormalization of the Landau growth rate or of the quasilinear diffusion coefficient is not necessarily related to the limit where the Landau growth time becomes large with respect to the time of spreading of the particle positions due to velocity diffusion

Nonquasilinear evolution of particle velocity in incoherent waves with random amplitudes

8 pages Elsevier styleThe one-dimensional motion of $N$ particles in the field of many incoherent waves is revisited numerically. When the wave complex amplitudes are independent, with a gaussian distribution, the quasilinear approximation is found to always overestimate transport and to become accurate in the limit of infinite resonance overlap

Notions de physique statistique

IntroductionPréliminaire : présentation de la solution d'un problèmeRappels de théorie des probabilitésEnsemble statistique quantiqueIllustration de la démarche statistique : réactions unimoléculairesRemarques sur le magnétismeSystème isolé : ensemble microcanoniqueSystème fermé : ensemble canoniqueTransition de phase : approche de champ moyenSystèmes ouverts : ensemble grand-canoniqueStatistiques quantiquesDEACours de master

Basic microscopic plasma physics unified and simplified by N-body classical mechanics

Debye shielding, collisional transport, Landau damping of Langmuir waves, and spontaneous emission of these waves are introduced, in typical plasma physics textbooks, in different chapters. This paper provides a compact unified introduction to these phenomena without appealing to fluid or kinetic models, but by using Newton's second law for a system of $N$ electrons in a periodic box with a neutralizing ionic background. A rigorous equation is derived for the electrostatic potential. Its linearization and a first smoothing reveal this potential to be the sum of the shielded Coulomb potentials of the individual particles. Smoothing this sum yields the classical Vlasovian expression including initial conditions in Landau contour calculations of Langmuir wave growth or damping. The theory is extended to accommodate a correct description of trapping or chaos due to Langmuir waves. In the linear regime, the amplitude of such a wave is found to be ruled by Landau growth or damping and by spontaneous emission. Using the shielded potential, the collisional diffusion coefficient is computed for the first time by a convergent expression including the correct calculation of deflections for all impact parameters. Shielding and collisional transport are found to be two related aspects of the repulsive deflections of electrons

On frequency and time domain models of traveling wave tubes

We discuss the envelope modulation assumption of frequency-domain models of traveling wave tubes (TWTs) and test its consistency with the Maxwell equations. We compare the predictions of usual frequency-domain models with those of a new time domain model of the TWT

Gaussian convergence for stochastic acceleration of N particles in the dense spectrum limit

19 pp.International audienceThe velocity of a passive particle in a one-dimensional wave field is shown to converge in law to a Wiener process, in the limit of a dense wave spectrum with independent complex amplitudes, where the random phases distribution is invariant modulo $\pi/2$ and the power spectrum expectation is uniform. The proof provides a full probabilistic foundation to the quasilinear approximation in this limit. The result extends to an arbitrary number of particles, founding the use of the ensemble picture for their behaviour in a single realization of the stochastic wave field