Direct path from microscopic mechanics to Debye shielding, Landau damping and wave-particle interaction
The derivation of Debye shielding and Landau damping from the N-body description of plasmas is performed directly by using Newton’s second law for the N-body system. This is done in a few steps with elementary calculations using standard tools of calculus and no probabilistic setting. Unexpectedly, Debye shielding is encountered together with Landau damping. This approach is shown to be justified in the one-dimensional case when the number of particles in a Debye sphere becomes large. The theory is extended to accommodate a correct description of trapping and chaos due to Langmuir waves. On top of their well-known production of collisional transport, the repulsive deflections of electrons are shown to produce shielding, in such a way that each particle is shielded by all other ones, while keeping in uninterrupted motion.
On the validity of quasilinear theory
whereζm is the complex envelope for wave m, with wavenumber km, phase velocityvm= ωm/km and pulsationωm = ωp √ 1+3k2 mλ 2 D. The couplingε = √ 2η/(1+η) is determined by the ratio η = nb/np of beam densitynb to plasma density np. Let ∆vm = |vm+1−vm−1|/2,u0 = min(vm), u1 = max(vm) ; ∆vspec= u1−u0 is the phase velocity width of the wave spectrum. The saturation of the beam–plasma system was first predicted theoretically [6, 4] by considering the wave–particle interaction as perturbative and ne glecting all mode couplings in (1)-(2), except for their effect on the space averaged distribution f unction f̄ . This leads to the quasilinear (QL) equations couplinḡf (t,v) and the waves power spectrum ψ(t,v)
Time evolution of large classical systems
We begin with some very general and elementary remarks about nonequilibrium statistical mechanics. We then establish our notation for discussing finite systems of classical point particles, construct the microcanonical ensemble, and sketch some of the relations between statistical mechanics and ergodic theory.