  The generalized second law states the total entropy of any closed system as
the universe cannot decrease if we include black hole entropy. From the point
of view of an asymptotic observer, a black hole can be described at late time
as an open system at fixed temperature which can radiate energy and entropy to
infinity. I argue that for massless free quantum fields propagating on a black
hole background, we can define a black hole dynamical free energy using
observables defined at future null infinity which decreases on successive cross
sections. The proof of this spontaneous evolution law is similar to Wall's
derivation of the generalized second law and relies on the monotonicity
properties of the relative entropy. I discuss first the simpler case of the
Schwarzschild background in which the grey body factor are neglected and show
that in this case the free energy only depends on the Bondi mass, the Hawking
temperature and the von Neumann entropy of the propagating quantum fields. Then
I argue that taking into account the grey body factors adds a new term to the
thermodynamic potential involving the number of particles detected at future
null infinity conjugated to a chemical potential. Finally, I discuss the case
of the Kerr black hole for which an angular momentum piece needs to be added to
the free energy.
