Thermodynamics of nonlinear charged Brans-Dicke-BTZ black holes

We studied the thermodynamic properties of novel Brans-Dicke (BD) black holes (BHs) coupled to power-Maxwell nonlinear electrodynamics (NLED) in a three-dimensional (3D) spacetime. The Jordan frame BD field equations are nonlinear and too complex for analytic solving. We removed this problem by using the conformal transformations (CT) which translate the BD action to the Einstein frame. In the Einstein frame, where the theory is the well-known Einstein-dilaton gravity, the field equations are decoupled and easier to solve. We solved the problem of mathematical indeterminacy, that appeared in the Einstein-power-Maxwell-dilaton (EPMd) theory, by using an exponential ansatz function. Then, by exactly solving the field equations, we introduced two novel classes of 3D EPMd BHs with unusual asymptotes. We calculated the conserved and thermodynamic quantities to confirm the validity of the first law of BH thermodynamics (FLT) and to analyze the thermal stability of the EPMd BHs. By applying the inverse CT, we introduced 3D BD-Power-Maxwell (BDPM) BHs corresponding to the Einstein frame counterpart. By using the plots, we showed that our Jordan frame solutions can produce horizon-less, one-horizon, and two-horizon BHs. We proved that thermodynamic quantities are the same in both conformally related frames and concluded they are conformal-invariant. Based on this symmetry property, the FLT remains valid for the BDPM BHs, and they exhibit similar stability properties as the EPMd ones.