The nonequilibrium back-reaction of Hawking radiation to a Schwarzschild black hole

We investigate the nonequilibrium back reaction on the Schwarzschild black hole from the radiation field. The back reactions are characterized by the membrane close to the black hole. When the membrane is thin, we found that larger temperature difference can lead to more significant negative surface tension, larger thermodynamic dissipation cost and back reaction in energy and entropy as well as larger black hole area. This may be relevant to the primordial black holes in early universe. Moreover, our nonequilibrium model can resolve the inconsistency issue of the black hole back reaction under zero mass limit in the equilibrium case. In the thick membrane case, the nonequilibrium back reaction is found to be more significant than that in the thin membrane case. The nonequilibrium temperature difference can increase the energy and entropy loss as well as the thermodynamic dissipation of the black hole and the membrane back reactions. The nonequilibrium dissipation cost characterized by the entropy production rate appears to be significant compared to the entropy rate radiated by the black hole under finite temperature difference. This may shed light on the black hole information paradox due to the information loss from the entropy production rate in the nonequilibirum cases. The nonequilibrium thermodynamic fluctuations can also reflect the effects of the back-reactions of the Hawking radiation on the evolution of a black hole.