  Horizons of black branes have an associated entropy current with non-negative
divergence. We compute this divergence in a late-time transseries expansion for
an inhomogeneous system evolving towards a maximally symmetric asymptotically
anti-de Sitter black brane. The horizon area equilibrates on half the
time-scale set by the dominant quasinormal mode and we find a simple analytic
expression for this evolution purely in terms of the background and the
quasinormal mode frequencies. This computation does not require a gradient
expansion and is thus non-perturbative in momenta. We generalize this to
include scalar and gauge field matter in any number of dimensions. Restricting
to homogeneous evolution we match and prove earlier numerical work showing that
the apparent horizon entropy saturates the area theorem, in that its time
derivative periodically vanishes. The same is true for spherically symmetric
evolution towards the Schwarzschild black hole.
