Quasilinear diffusion in action space

The second-order, angle-averaged evolution of a galactic distribution function caused by weak fluctuating gravitational perturbations. In action-angle variables, the first-order response to a Fourier series harmonic feeds a second-order flux proportional to $\mathbf{k}(\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0)$ and to the perturbation's power. Orbital resonances select where transport is strongest; a finite pulse produces Gaussian resonance broadening when its time envelope is a Gaussian function. The general flux retains time memory until a further short-memory or pulse-integrated approximation is made.

For a perturbation with amplitude envelope $e^{-(t-t_0{)}^2/(2{\tau }^2)}$, the squared Fourier transform is proportional to $e^{-{\omega }^2{\tau }^2}$. Hence pulse-integrated transport weights orbital detuning with a normalized Gaussian function of width $\Gamma =(\sqrt{2}\tau {)}^{-1}$. A longer pulse selects a narrower band of orbital resonances. This spectral broadening does not by itself assume stochastic temporal forcing.