Quasilinear approximation

A weak-fluctuation closure retaining the averaged quadratic fluctuation flux while linearizing the fluctuating kinetic equation about a slowly evolving mean phase-space distribution function. One solves the fast linear response with the mean frozen, then inserts its covariance into the slow averaged equation. A short-memory stationary limit can yield quasilinear diffusion, but finite-time responses retain memory. Self-consistent collective fields enter through the dielectric function in homogeneous electrostatic problems.

The diffusive evolution of an averaged phase-space distribution function caused by weak fluctuating fields in the quasilinear approximation. In a stationary homogeneous one-dimensional electrostatic problem, if the potential's temporal power spectral density is $S_{\phi ,k}(\omega )$ with covariance $\int S_{\phi ,k}(\omega )e^{-i\omega \tau }d\omega /(2\pi )$, thenThis follows from the real part of the one-sided time correlation along an unperturbed particle trajectory. Both signs of the spatial wavenumber are included. The action-space counterpart for regular stellar orbits is quasilinear diffusion in action space.

Random changes of particle velocity described by a diffusion equation in velocity space. For constant $D$, $⟨(\Delta v{)}^2⟩=2D\Delta t$. With zero boundary flux it conserves number. For a number-normalized one-dimensional distribution, its kinetic energy production is $d\mathcal{K}/dt=-m\int vD(v)f^{\prime }(v)dv$, which reduces to $mnD$ for constant $D$. Variable $D$ must remain inside the derivative.