Galactic bar resonances with diffusion: an analytic model with implications for bar-dark matter halo dynamical friction

The secular evolution of disk galaxies over cosmic time is largely driven by resonances between the orbits of 'particles' (e.g. stars or dark matter) and the rotation of non-axisymmetric features (e.g. spiral arms or a bar). Such resonances are also often invoked to explain present-day kinematic and photometric features observed in the Milky Way and external galaxies. In simplified cases these resonant interactions are well understood: for instance, the secular dynamics of a test particle trapped near a resonance of a steadily rotating bar is easily analyzed using the angle-action tools pioneered by Binney, Monari and others. However, their treatments don't address the stochasticity and messiness of real galaxies - effects which have, with few exceptions, been previously captured only in complex N-body simulations. In this paper we propose a simple kinetic equation to describe the distribution function of particles near an orbital resonance with a rigidly rotating bar, using the pendulum approximation and allowing for diffusion of the particles' slow actions. We solve this kinetic equation for various values of the diffusion strength $\Delta$. We then apply our theory to the calculation of the dynamical friction torque felt by a bar embedded in a spherical dark matter halo. For $\Delta = 0$ we recover the classic result of Tremaine & Weinberg that the friction vanishes in the time-asymptotic (phase-mixed) limit, whereas for $\Delta > 0$ we find that diffusion suppresses phase mixing, leading to a finite negative torque, suggesting that real galactic bars always decelerate.