FIG. 7: 10 − $^{6}$( ∂ $^{2}$M/∂S$^{2}$)$_{Q}$ (solid line) and T (dashed line) versus α for b = 0 . 2, n = 4, l = 1 and r$_{+}$ = 0 . 001. FIG. 8: 10 − $^{5}$( ∂ $^{2}$M/∂S$^{2}$)$_{Q}$ (solid line) and T (dashed line) versus α for b = 0 . 2, n = 4, l = 1 and r$_{+}$ = 0 . 01. of a thermodynamic system with respect to small variations of the thermodynamic coordi- nates is usually performed by analyzing the behavior of the entropy S ( M, Q ) around the equilibrium. The local stability in any ensemble requires that S ( M, Q ) be a convex function of the extensive variables or its Legendre transformation must be a concave function of the intensive variables. The stability can also be studied by the behavior of the energy M ( S, Q ) which should be a convex function of its extensive variable. Thus, the local stability can in principle be carried out by finding the determinant of the Hessian matrix of M ( S, Q ) with respect to its extensive variables X$_{i}$ , H M X$_{i}$X$_{j}$ = [ ∂ $^{2}$M/∂X$_{i}$ ∂X$_{j}$ ] [33, 37]. In our case the mass M is a function of entropy and charge. The number of thermodynamic variables depends 14 k = –1 –0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 alpha k = 0 –0.1 –0.05 0 0.05 0.1 1 2 3 4 5 alpha k = –1 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 alpha k = 0 –0.6 –0.4 –0.2 0 0.2 0.4 1 1.5 2 2.5 3 3.5 alpha