The celebrated prediction of black hole entropy and Hawking radiation made black hole thermodynamics an interesting field in cosmology and high energy physics. Black hole, as a solution to Einstein field equation, behaves like a thermodynamic system. One of such thermodynamic properties is the phase transition between locally stable and unstable black hole, investigated by Davies for the first time in 1977. An important question about this phase transition is the order of transition. The order of phase transition shows that whether entropy and volume of a system change continuously or discontinuously in the transition. If these changes are discontinuous then the transition is of first order, otherwise it is called a higher order or continuous transition. This kind of In this thesis we investigate thermodynamic phase transition for a large We study some useful definitions in ensemble theory of black holes throughout this dissertation. In canonical ensemble, extensive parameters such as electrical charge or angular momentum are supposed to be fixed at the boundary of black hole. In grand canonical ensemble extensive parameters can be exchanged between black hole and its environment. We also investigate Kerr-Newman black hole which has three degrees of freedom. In this case there are also two mixed ensembles which are defined by taking one extensive parameter and one intensive parameter to be fixed at the horizon, while their conjugate quantities can be changed. The discontinuity of specific heat reflects the possibility of the second order phase transition in canonical and two mixed ensembles. However there are differences between the second order phase transition in black holes and ordinary thermodynamic systems for which the discontinuity of specific heat is not an infinite one. Usual Ehrenfest’s equations are written for systems with two degrees of freedom, so they are helpless for Kerr-Newman black hole with three degrees of freedom. In this thesis we generalize Ehrenfest’s equations. We find seven new equations with six new parameters that diverge at the same point in which the heat capacity diverges. We show that all generalized Ehrenfest’s equations are satisfied at the divergent point of specific heat in canonical and two mixed ensembles. So Kerr-Newman black hole undergoes a second order phase transition in these ensembles. Keywords: Thermodynamics, Black hole, Phase transition, Klapeyron, Ehrenfest, Stability of black holes, Landau theory, Hawking-Page phase transition.
