Evolution of binaries under stochastic perturbations

We develop a general Fokker-Planck framework describing the dynamical evolution of Keplerian binaries subjected to stochastic perturbations. The formalism provides an algorithmic way to obtain the Fokker-Planck drift and diffusion coefficients of any set of orbital variables given the statistics of the perturbations. We apply the method to three physically distinct regimes: adiabatic tidal perturbations, white-noise tidal perturbations, and impulsive encounters with a third body of arbitrary density profile. In each regime we provide explicit drift and diffusion coefficients for all six orbital elements, derive the associated evolution timescale, and obtain analytic steady-state distribution functions. Our results extend previous treatments by including the evolution of the binary's orientation, retaining the complete tensor structure of tidal correlators, treating non-pointlike perturbers, and resolving the exact geometry of impulsive encounters. The latter correction leads to a steady-state eccentricity distribution that is slightly sub-thermal. We also show how these equations can be applied directly in several astrophysical scenarios, including binaries perturbed by dark matter subhaloes, ultralight dark matter, and the interstellar medium. This work delivers both a complete mathematical framework and a practical toolkit for stochastic binary evolution, providing ready-to-evaluate equations to be applied directly to binary population data.