Fluctuation‐dissipation‐dispersion relations for a nonuniform plasma

Abstract

Using the Langevin approach and the multiscale technique, a kinetic theory of the time and space nonlocal fluctuations in the collisional plasma is constructed. In local equilibrium a generalized version of the CallenWelton theorem is derived. It is shown that not only the dissipation but also the time and space derivatives of the dispersion determine the amplitude and the width of the spectrum lines of the electrostatic field fluctuations, as well as the form factor. There appear significant differences with respect to the non-uniform plasma. In the kinetic regime the form factor is more sensible to space gradient than the spectral function of the electrostatic field fluctuations. As a result of the inhomogeneity, these proprieties became asymmetric with respect to the inversion of the frequency sign. The differences in amplitude of peaks could become a new tool to diagnose slow time and space variations in the plasma.Fluctuations play an essential role in the plasma diagnostics. Indeed, plasma parameters such as temperature, mean velocity, density can be determined by incoherent (Thomson) scattering diagnostics [1], i.e. by the proper interpretation of data obtained from the scattering of a given electromagnetic field interacting with the system. The key point of interpreting them is the knowledge of the intensity of the dielectric function fluctuations or equally of the electron form factor (δn e δn e ) ω,k . Here ω and k are respectively the frequency and wavevector of the autocorrelations. Due to the Poisson equation the electron form factor in the spatially homogeneous system is directly linked to the electrostatic field fluctuations, which have been the object of active study since the early 1960s [1]. In the thermodynamic equilibrium, the electrostatic field fluctuations satisfy the famous Callen-Welton fluctuation-dissipation theorem [2]:linking their intensity to the imaginary (dissipative) part of the dielectric function ε(ω, k), and the temperature Θ in energy units. The spectral function has peaks, corresponding to proper plasma frequencies. The matter becomes more tricky in the non-equilibrium case. When the state of the plasma is given by Maxwell distributions characterized by different constant temperatures and velocities per species (Θ a , V a ; a = e, i), it is generally admitted that the spectral function of the electron density for a two component system takes the form [1]:(δn e δn e ) ωk =2 . k D is the Debye number and χ a (a = e, i) is the complex dielectric susceptibility of the a-th component. This formula has been extensively used to interpret the scattering data mentioned above. We have indeed shown [3], that, in the collisional regime equations this formula should be revisited. We stressed the fact that a kinetic approach should be taken. Introducing fluctuations by the Langevin method, we have elaborated a "revisited" Callen-Welton formula containing new terms explicitly displaying dissipative non equilibrium contributions. It is however not evident that the plasma parameters -tempera...