Linear response, dynamical friction and the fluctuation-dissipation theorem in stellar dynamics

We apply linear response theory to a general, inhomogeneous, stationary stellar system, with particular emphasis on dissipative processes analogous to Landau damping. Assuming only that the response is causal, we show that the irreversible work done by an external perturber is described by the anti-Hermitian part of a linear response operator, and damping of collective modes is described by the anti-Hermitian part of a related polarization operator. We derive an exact formal expression for the response operator, which is the classical analog of a well-known result in quantum statistical physics. When the self-gravity of the response can be ignored, and the ensemble-averaged gravitational potential is integrable, the expressions for the mode energy, damping rate, and polarization operator reduce to well-known formulae derived from perturbation theory in action-angle variables. In this approximation, dissipation occurs only via resonant interaction with stellar orbits or collective modes. For stellar systems in thermal equilibrium, the anti-Hermitian part of the response operator is directly related to the correlation function of the fluctuations. Thus dissipative properties of the system are completely determined by the spectrum of density fluctuations—the fluctuation-dissipation theorem. In particular, we express the coefficient of dynamical friction for an orbiting test particle in terms of the fluctuation spectrum; this reduces to the known Chandrasekhar formula in the restrictive case of an infinite homogeneous system with a Maxwellian velocity distribution.