Critical behaviour of Lifshitz dilaton black holes

Abstract

Till now, critical behaviour of Lifshitz black holes, in an extended $P-v$ space, has not been studied, because it is impossible to find an analytical equation of state, $P=P(v,T)$, for an arbitrary Lifshitz exponent $z$. In this paper, we adopt a new approach toward thermodynamic phase space and successfully explore the critical behaviour of $(n+1)$-dimensional Lifshitz dilaton black holes. For this purpose, we write down the equation of state as $Q^s=Q^s(T,\Psi)$ with $\Psi=\left({\partial M}/{\partial Q^{s} }\right)_{S,P}$ is the conjugate of $Q^s$ and construct Smarr relation based on this new phase space as $ M=M(S,Q^{s},P)$, where $s=2p/(2p-1)$ with $p$ is the power of the power-law Maxwell Lagrangian. We justify such a choice mathematically and show that with this new phase space, the system admits the critical behaviour and resembles the Van der Waals fluid system when the cosmological constant (pressure) is treated as a fixed parameter, while the charge of the system varies. We obtain Gibbs free energy of the system and find swallow tail shape in Gibbs diagrams which represents the first order phase transition. Finally, we calculate the critical exponents and show that although thermodynamic quantities depend on the metric parameters such as $z$ , $p$ and $n$, the critical exponents are the same as Van der Walls fluid-gas system. This alternative viewpoint toward phase space of lifshitz dilaton black hole can be understood easily since one can imagine such a change for a given single black hole i.e., acquiring charge which induces the phase transition. Our results further support the viewpoint suggested in \cite{Dehy}.