Thermodynamics of nonlinear charged Lifshitz black branes with hyperscaling violation

In this paper, we investigate the thermodynamics of hyperscaling-violating Lifshitz black branes in the presence of a nonlinear massless electromagnetic field. We, first, obtain analytic nonlinear charged black brane solutions with a hyperscaling-violating factor in dilaton gravity and give the condition on the parameters of the metric for having black brane solutions. Second, we introduce the appropriate finite action in the grand canonical and canonical ensembles for a nonlinear electromagnetic field. Next, by generalizing the counterterm method for the asymptotic Lifshitz spacetimes with a hyperscaling-violating factor, we calculate the energy density of our solutions. Furthermore, we present a relation between the energy density and the thermodynamic quantities, electric potential, charge density, temperature and entropy density. This relation is the generalization of the Smarr formula for anti–de Sitter black branes and charged Lifshitz solutions. Finally, we perform a stability analysis in both the canonical and grand canonical ensembles. We show that the nonlinearity of the electromagnetic field can make the solutions unstable in the grand canonical ensemble.

Figure 1

Temperature (red, dotted) and $Ϝ$ (blue, line) as functions of the horizon radius for $q=0.6$, $d=3$, $z=3$ $s=4$ and $\theta =2$.