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