Black Hole Evaporation and Nonequilibrium Thermodynamics for a Radiation   Field
  When a black hole is put in an "empty" space (zero temperature space) on which there is no matter except the matter of the Hawking radiation (Hawking field), then an outgoing energy flow from the black hole into the empty space exists. By the way, an equilibrium between two arbitrary systems can not allow the existence of an energy (heat) flow from one system to another. Consequently, in the case of a black hole evaporation in the empty space, the Hawking field should be in a nonequilibrium state. Hence the total behaviour of the evaporation, for example the time evolution of the total entropy, should be analysed with a nonequilibrium thermodynamics for the Hawking field. This manuscript explains briefly the way of constructing a nonequilibrium thermodynamic theory for a radiation field, and apply it to a simplified model of a black hole evaporation to calculate the time evolution of the total entropy. 
Hiromi Saida