I am a little confused over something, so perhaps someone can tell me which one of the following statements is incorrect. (I don't know a lot about this, so forgive my ignorance). Suppose that we have a closed system of some size, and we start in a state where we have a black hole at the center. So there is some entropy S1 at our initial condition. Now we let the system evolve and the black hole starts radiating and losing mass. At some point very far in the future the black hole radiates completely away and we are left with a lot of particles that are distributed (uniformly?) throughout our system. Our final condition here has some entropy S2. The second law of thermodynamics says that S2 >= S1. However, it was my understanding that in the presence of gravity a large number of particles uniformly distributed will have a lower entropy that when they are clumped together (like our original black hole). So it seems like S1 < S2. Or are so many particles created during the radiation that S2 > S1?
It looks like Zurek figured out the answer some time ago. The answer is that the end entropy is ~4/3 times the initial entropy. For anyone else interested in this, see W. H. Zurek, Phys. Rev. Lett. 49 1683 (1982).