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Thermodynamics of rotating black branes in $(n+1)$-dimensional Einstein-Born-Infeld-dilaton gravity

In this paper, we construct a new class of charged, rotating solutions of $% (n+1)$-dimensional Einstein-Born-Infeld-dilaton gravity with Liouville-type potentials and investigate their properties. These solutions are neither asymptotically flat nor (anti)-de Sitter. We find that these solutions can represent black brane, with inner and outer event horizons, an extreme black brane or a naked singularity provided the parameters of the solutions are chosen suitably. We also compute temperature, entropy, charge, electric potential, mass and angular momentum of the black brane solutions, and show that these quantities satisfy the first law of thermodynamics. We find that the conserved quantities are independent of the Born-Infeld parameter $β$, while they depend on the dilaton coupling constant $α$. We also find the total mass of the black brane with infinite boundary as a function of the entropy, the angular momenta and the charge and perform a stability analysis by computing the heat capacity in the canonical ensemble. We find that the system is thermally stable for $α\leq 1$ independent of the values of the charge and Born-Infeld parameters, while for $α> 1$ the system has an unstable phase. In the latter case, the solutions are stable provided $α\leq α_{\max}$ and $β\geq β_{\min}$, where $% α_{\max}$ and $β_{\min}$ depend on the charge and the dimensionality of the spacetime. That is the solutions are unstable for highly nonlinear electromagnetic field or when the dilaton coupling constant is large.