Stellar dynamics in the periodic cube

We use the problem of dynamical friction within periodic cube to illustrate application perturbation theory in stellar dynamics, testing its predictions against measurements from $N$-body simulation. Our development is based on explicitly time-dependent Volterra integral equation for cube's linear response, which avoids subtleties encountered analyses complex frequency. obtain an expression self-consistent response steady stirring by external perturber. From this we show how familiar Chandrasekhar formula and construct elementary derivation Lenard--Balescu secular quasilinear evolution isolated composed $N$ equal-mass stars. present alternative (real-frequency) van Kampen modes decompose any into a superposition such modes.