Thermodynamic critical and geometrical properties of charged BTZ black hole

The heat capacities and the electric capacitances of charged Ba\~nados-Teitelboim-Zanelli (BTZ) black hole are first calculated. By using the equilibrium fluctuation theory of thermodynamics the second-order moments in three different ensembles are obtained, and it is found that in the microcanonical ensemble the extremal charged BTZ black hole is a critical point of the second-order phase transition. The critical exponents associated with some response coefficients satisfy the scaling law of the first kind and the effective spatial dimension is determined to be one from the scaling law of the second kind. The Ricci curvature scalar associated with the Ruppeiner thermodynamic metric is calculated, which suggests also that the effective spatial dimension of the charged BTZ black hole is one.

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