White Dwarf Natal Kicks as Velocity-Space Random Walks. I. Fokker-Planck Theory

White dwarfs are the most common end state of stellar evolution, including that of our own Sun, making their formation critical to completing a more comprehensive understanding of stellar evolution and even determining the eventual fate of the solar system. Wide binaries containing white dwarfs offer a powerful probe: their components evolve largely independently, yet stellar mass loss and velocity impulses (natal kicks) on the white dwarf progenitor leave measurable imprints on orbital properties. Previous population studies typically assume that each white dwarf receives only a single natal kick as it forms. While convenient, this simplification conflicts with simulations finding the formation process to be episodic. Therefore, a more realistic picture is that white-dwarf progenitors experience multiple stochastic kicks over their evolution. Here, we present a Fokker-Planck model for the evolution of binary orbital parameter distributions subject to many, randomly-oriented mass-loss-driven kicks. This model recovers key features of white-dwarf binary observations, including a-2 power law tail in the probability distribution of separations, and the thermalization of binary eccentricities beyond separations of 1000 au. We additionally investigate how white-dwarf kicks impact a myriad of other processes, such as binary dissolution, planetary system evolution, and even the production of interstellar objects.