  If a black hole starts in a pure quantum state and evaporates completely by a
unitary process, the von Neumann entropy of the Hawking radiation initially
increases and then decreases back to zero when the black hole has disappeared.
Here numerical results are given for an approximation to the time dependence of
the radiation entropy under an assumption of fast scrambling, for large
nonrotating black holes that emit essentially only photons and gravitons. The
maximum of the von Neumann entropy then occurs after about 53.81% of the
evaporation time, when the black hole has lost about 40.25% of its original
Bekenstein-Hawking (BH) entropy (an upper bound for its von Neumann entropy)
and then has a BH entropy that equals the entropy in the radiation, which is
about 59.75% of the original BH entropy 4 pi M_0^2, or about 7.509 M_0^2
\approx 6.268 x 10^{76}(M_0/M_sun)^2, using my 1976 calculations that the
photon and graviton emission process into empty space gives about 1.4847 times
the BH entropy loss of the black hole. Results are also given for black holes
in initially impure states. If the black hole starts in a maximally mixed
state, the von Neumann entropy of the Hawking radiation increases from zero up
to a maximum of about 119.51% of the original BH entropy, or about 15.018 M_0^2
\approx 1.254 x 10^{77}(M_0/M_sun)^2, and then decreases back down to 4 pi
M_0^2 = 1.049 x 10^{77}(M_0/M_sun)^2.
