**Abstract** : We 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 as
Bernstein 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 nonlinear
Schrödinger equation for fluctuating macroscopic quantities, like the density of trapped particles.
Naïvely, a statistical time dependant perturbative treatment of the relaxation of excitations comes
from self-scattering. Dyson’s equation accounts for the renormalised bulk plasma frequency. This
setting also predicts a Lamb shift for the only remaining ground state energy mode. Landau
damping is retrieved when one estimates the energy spectrum from local equilibrium, which shows
a 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 plasma
perturbations. 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 statistical
treatment is traced back to the presence of intrinsic disorder from the wild oscillations in phase
space. Precisely, a minimal mean field model is written for the coarse grained dynamics. It is
instructive to draw an illustration from the idea of a hierarchy of KAM tori and the emergence of
frustration 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 is
renormalised chaotically. Thus, defining a local phase for the spatially inhomogeneous oscillations
of macroscopic quantities, it is possible to «geometrize» their symplectic dynamics, thereby
justifying the initially intuitive «quantum statistical» treatment of the Schrödinger equation.
For driven systems, the chaotic parameter further forces large deviations on the dynamics, with
non-Maxwellian stationary distributions, or strong turbulence. Therefore, since at large scales
exchange entropy with bulk plasma and (KS-)entropy production rates are high, a comparison with
coupled chaotic map lattices for coarse grained observables is qualitatively worthy. This implies a
rich structure of phase space diagram, where many transitions are to be expected between stable
or metastable states (phases), related to QSSs (quasi stationary states).