Bridging Between Statistical Mechanics and Black Hole Evolution: Theoretical Formalism

Black holes in the Universe span nearly ten orders of magnitude in mass over a large cosmic timescale, dating back to at least the Universe's early history, when it was only a few hundred million years old. The formation and evolution of these objects remain a mystery because no direct probe can trace each black hole across the full mass range and cosmic time. This work shows, for the first time, a statistical mechanical description of black hole evolution. I show that though each black hole and its evolution are governed by stochastic processes, one can make a macroscopic description of the distribution function of black holes in the Universe, and its evolution over the entire mass range and cosmic time can be expressed in terms of the Kramers-Moyal expansion. This bridging between statistical mechanics and black hole evolution in terms of the Kramers-Moyal expansion enables the discovery of the underlying physical processes that play a role in the evolution of black holes. In the future, the application of this technique to data accessible from different messengers, such as gravitational waves and electromagnetic waves, which probe black holes in the Universe, can lead to a data-driven understanding of the underlying physical processes by this statistical mechanics technique despite the ignorance of the underlying microscopic stochastic processes.