  Recently, a class of stationary black hole solutions with non-killing horizon
in the asymptotic AdS bulk space was constructed to describe the far from
equilibrium heat transport and particle transport from the boundary black holes
via AdS/CFT correspondence. In this study, we calculate the spectrum of Hawking
radiation of the black funnel solution. Our results indicate that the spectrum
and the temperatures as well as the chemical potentials of the non-equilibrium
black funnel do depend on one of the spatial coordinates. This is different
from the equilibrium black holes with killing horizon, where the temperatures
are uniform. Therefore, the black hole with non-killing horizon can be overall
in non-equilibrium steady state while the Hawking temperature of the black
funnel can be viewed as the local temperature and the corresponding Hawking
radiation can be regarded as being in the local equilibrium with the horizon of
the black funnel. By AdS/CFT, we discuss some possible implications of our
results of local Hawking temperature for the non-equilibrium thermodynamics of
dual conformal field theory. We further discuss the nonequilibrium
thermodynamics of the black funnel, where the first law can be formulated as
the entropy production rate being equal to the sum of the changes of the
entropies from the system (black funnel) and environments while the second law
is given by the entropy production being larger than or equal to zero. We found
the time arrow emerged from the nonequilibrium black hole heat and particle
transport dissipation. We also discuss how the nonequilibrium dissipation may
influence the evaporation process of the black funnel.
