  Understanding distributions of black holes is crucial to both astrophysics
and quantum gravity. Studying astrophysical population statistics has even been
suggested as a channel to constrain black hole formation from the quantum
vacuum. Here we propose a Gedankenexperiment to show that the non-linear
properties of binary mergers (simulated with accurate surrogate models)
generate an attractor in the space of distributions. Our results show that the
joint distribution of spin magnitude and fractional mass loss evolves to a
fixed point, converging in a few generations. The features of this fixed point
distribution do not depend on the choice of initial distribution. Since a black
hole merger is irreversible it produces entropy - possibly the largest source
of entropy in the universe. The fixed-point distributions are neither
isothermal nor isentropic, and initially thermodynamic states evolve away from
thermality. We finally evaluate the specific entropy production rate per merger
from initially thermal and non-thermal distributions, which converges to a
constant.
