https://hal.inria.fr/hal-01322853Attuel, GuillaumeGuillaumeAttuelGeoStat - Geometry and Statistics in acquisition data - Inria Bordeaux - Sud-Ouest - Inria - Institut National de Recherche en Informatique et en AutomatiqueSurprising "quantum statistical" treatment of Landau damping: discussion on simple models.HAL CCSD2016Statistical mechanics long range interactions Landau damping path integrals desynchronization[PHYS.COND.CM-SM] Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech][PHYS.COND.CM-DS-NN] Physics [physics]/Condensed Matter [cond-mat]/Disordered Systems and Neural Networks [cond-mat.dis-nn][PHYS.QPHY] Physics [physics]/Quantum Physics [quant-ph]Yahia, H.2016-06-07 15:16:152022-01-20 16:17:302016-06-07 15:49:51enConference papers1We elaborate on quite a surprising mapping between plasma excitation Landau damping to a«quantum statistical» perturbative treatment of Schrödinger equation in a nonlinear potential.Firstly, the setting for this mapping is to assume spatially inhomogeneous stable modes, such asBernstein Green Kruskal modes. Bulk plasma frequency is then locally slightly altered by those.From a low frequency approximation of Bohm and Gross dispersion relation, we write a nonlinearSchrödinger equation for fluctuating macroscopic quantities, like the density of trapped particles.Naïvely, a statistical time dependant perturbative treatment of the relaxation of excitations comesfrom self-scattering. Dyson’s equation accounts for the renormalised bulk plasma frequency. Thissetting also predicts a Lamb shift for the only remaining ground state energy mode. Landaudamping is retrieved when one estimates the energy spectrum from local equilibrium, which showsa condensation at Debye’s wavelength. Or in other words, a gap has opened.It is in full agreement with the fluctuation-dissipation theorem and holds for all time.This mapping seems to be a natural continuum limit for a statistical treatment of bulk plasmaperturbations. It can be extended to the non perturbative case of Lynden-Bell violent relaxation.Secondly, wediscuss the justification of renormalisation in this context. Naturally, the statisticaltreatment is traced back to the presence of intrinsic disorder from the wild oscillations in phasespace. Precisely, a minimal mean field model is written for the coarse grained dynamics. It isinstructive to draw an illustration from the idea of a hierarchy of KAM tori and the emergence offrustration for the oscillations of trapped particles. Based on ideas by D. Escande and S. McKay,we arrive at an effective coupling between neighbouring clusters of trapped particles which isrenormalised chaotically. Thus, defining a local phase for the spatially inhomogeneous oscillationsof macroscopic quantities, it is possible to «geometrize» their symplectic dynamics, therebyjustifying the initially intuitive «quantum statistical» treatment of the Schrödinger equation.For driven systems, the chaotic parameter further forces large deviations on the dynamics, withnon-Maxwellian stationary distributions, or strong turbulence. Therefore, since at large scalesexchange entropy with bulk plasma and (KS-)entropy production rates are high, a comparison withcoupled chaotic map lattices for coarse grained observables is qualitatively worthy. This implies arich structure of phase space diagram, where many transitions are to be expected between stableor metastable states (phases), related to QSSs (quasi stationary states).