  When a black hole is in an empty space on which there is no matter field
except that of the Hawking radiation (Hawking field), then the black hole
evaporates and the entropy of the black hole decreases. The generalised second
law guarantees the increase of the total entropy of the whole system which
consists of the black hole and the Hawking field. That is, the increase of the
entropy of the Hawking field is faster than the decrease of the black hole
entropy. In naive sense, one may expect that the entropy increase of the
Hawking field is due to the self-interaction among the composite particles of
the Hawking field, and that the "self"-relaxation of the Hawking field results
in the entropy increase. Then, when one consider a non-self-interacting matter
field as the Hawking field, it is obvious that the self-relaxation does not
take place, and one may think that the total entropy does not increase.
However, using nonequilibrium thermodynamics which has been developed recently,
we find for the non-self-interacting Hawking field that the rate of entropy
increase of the Hawking field (the entropy emission rate by the black hole)
grows faster than the rate of entropy decrease of the black hole along the
black hole evaporation in the empty space. The origin of the entropy increase
of the Hawking field is the increase of the black hole temperature. Hence an
understanding of the generalised second law in the context of the
nonequilibrium thermodynamics is suggested; even if the self-relaxation of the
Hawking field does not take place, the temperature increase of the black hole
during the evaporation process causes the entropy increase of the Hawking field
to result in the increase of the total entropy.
