Generalized Ehrenfest's equations and phase transition in black holes

In this paper, we generalize Ehrenfest's equations to systems having two work terms, i.e. systems with three degrees of freedom. For black holes with two work terms we obtain nine equations instead of two to be satisfied at the critical point of a second-order phase transition. We finally generalize this method to a system with an arbitrary number of degrees of freedom and found there is N(N+1)(2)/2 equations to be satisfied at the point of a second-order phase transition where N is number of work terms in the first law of thermodynamics.