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

We apply linear response theory to a general, inhomogeneous, stationary ste llar system, with particular emphasis on dissipative processes analogous to Landau damping. Assuming only that the response is causal, we show that th e irreversible work done by an external perturber is described by the anti- Hermitian part of a linear response operator, and damping of collective mod es is described by the anti-Hermitian part of a related polarization operat or. We derive an exact formal expression for the response operator, which i s the classical analogue of a well-known result in quantum statistical phys ics. When the self-gravity of the response can be ignored, and the Hamilton ian corresponding to the ensemble-averaged gravitational potential is integ rable, the expressions for the mode energy, damping rate and polarization o perator reduce to well-known formulae derived from perturbation theory in a ction-angle variables. In this approximation, dissipation occurs only via r esonant interaction with stellar orbits or collective modes. For stellar sy stems in thermal equilibrium, the anti-Hermitian part of the response opera tor is directly related to the correlation function of the fluctuations. Th us dissipative properties of the system are completely determined by the sp ectrum of density fluctuations - the fluctuation dissipation theorem. In pa rticular, we express the coefficient of dynamical friction for an orbiting test particle in terms of the fluctuation spectrum; this reduces to the kno wn Chandrasekhar formula in the restrictive case of an infinite homogeneous system with a Maxwellian velocity distribution.