  The secular evolution of disk galaxies is largely driven by resonances between the orbits of 'particles' (stars or dark matter) and the rotation of non-axisymmetric features (spiral arms or a bar). Such resonances may also explain kinematic and photometric features observed in the Milky Way and external galaxies. In simplified cases, these resonant interactions are well understood: for instance, the 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, such treatments do not address the stochasticity and messiness inherent to real galaxies - effects which have, with few exceptions, been previously explored only with complex N-body simulations. In this paper, we propose a simple kinetic equation describing the distribution function of particles near an orbital resonance with a rigidly rotating bar, allowing for diffusion of the particles' slow actions. We solve this equation for various values of the dimensionless diffusion strength $\Delta$, and then apply our theory to the calculation of bar-halo dynamical friction. For $\Delta = 0$ we recover the classic result of Tremaine & Weinberg that friction ultimately vanishes, owing to the phase-mixing of resonant orbits. However, for $\Delta > 0$ we find that diffusion suppresses phase-mixing, leading to a finite torque. Our results suggest that stochasticity - be it physical or numerical - tends to increase bar-halo friction, and that bars in cosmological simulations might experience significant artificial slowdown, even if the numerical two-body relaxation time is much longer than a Hubble time. 
Galactic bar resonances with diffusion: an analytic model with   implications for bar-dark matter halo dynamical friction