Steady-state Black Holes Of The Two Phase Transitions

The position of the second order phase transition points for the stationary black holes has been argued by many authors recently. Some people draw a conclusion that the second order phase transition point in the stationary black hole lies in where the heat capacity tend to infinity by means of the thermodynamical equilibrium fluctuations. But the other people obtain a different conclusion that the second order phase transition takes place when r+ - r- by using of the thermodynamical non-equilibrium fluctuations. The controversy has been not resolved for a long time. In this paper, making use of both the thermodynamical equilibrium and the non-equilibrium fluctuations, we ascertain the position of the second order phase transition points in the stationary black holes, and discuss their physic meanings.We first introduce the second order phase transition theory in the general thermodynamical system, and make a brief summary of the about investigations of the second order phase transitions in stationary black holes in the recent years. Then we expatiate on the methods of investigating phase transition, in particularly, and expound the theories of equilibrium and non-equilibrium fluctuations. After that, both the equilibrium and the non-equilibrium fluctuations of thermodynamical quantities in Schwarzschild black hole, Reissner-Nordstrom black hole, charged dilaton black hole, and Kerr-Newman black hole are calculated. We find (1) The second order phase transitions don't occurin Schwarzschild black hole and in charged dilaton black hole when a = 1; (2) The second order phase transitions take place at the points where the heat capacities are diverged in the Reissner-Nordstrom black hole, charged dilaton black hole when a 1, and Kerr-Newman black hole. At the phase transition points, both the thermodynamical equilibrium and non-equilibrium fluctuations in these black holes approach infinite; (3) Because both of the thermodynamical fluctuations in all stationary black holes listed above approach zero, the second order phase transitions dosen't occur when r+ r_.In summary, whether making use of the thermodynamical equilibrium fluctuations or the non-equilibrium fluctuations, we find that the second order phase transition points lie in where the heat capacities approach infinite, rather than where r+ r_ in the stationary black holes. Thus the question which has been controverted for a long time is settled.