# Reentrant phase transition of Born-Infeld-dilaton black holes

###### Abstract

In this paper, we consider the cosmological constant as pressure and study the reentrant phase transition of -dimensional Born-Infeld-dilaton black holes in the canonical ensemble of extended phase space. We show that these black holes enjoy a zeroth order intermediate/small black hole phase transition, and also, there is a first order phase transition between small and large black holes.

## I Introduction

It was shown that black holes can be considered as thermodynamical systems Davies (); Davies1989 (); Wald () with typical temperature Hawking () and entropy Bekenstein () which satisfy the first law of black hole thermodynamics Bardeen (). It has also been found that they can behave like ordinary thermodynamical systems and undergo the phase transition HP (). A more interesting case was observed when one considers a correspondence of variables for charged black holes which results into the van der Waals like phase transition Chamblin (); ChamblinEmparan (); Shen (). In recent years, the idea of considering the cosmological constant as thermodynamical pressure has attracted a lot of attention in black hole thermodynamics Caldarelli (); Kastor ()

(1) |

where the thermodynamical quantity conjugate to is the thermodynamical volume

(2) |

in which “” stands for “residual extensive parameters”. The motivation comes from the fact that in some fundamental theories there are several physical constants, such as Yukawa coupling, gauge coupling constants, and Newton’s constant that are not fixed values. On the other hand, the cosmological constant stands by pressure side in Tolman–Oppenheimer–Volkoff equation which shows the cosmological constant can be considered as thermodynamical pressure. Considering as pressure of system leads to a van der Waals like small/large black holes (SBH/LBH) phase transition which has been investigated by so many authors (for instance see incomplete list KubiznakMann (); Banerjee (); Mirza (); Mo (); Zou (); Xu (); HendiFaizal (); Mandal (); HendiPV (); MassiveYM () and references therein).

The reentrant phase transition (RPT) can occur in a thermodynamical system whenever a monotonic change of any thermodynamic variable leads into more than one phase transition so that the final state is macroscopically similar to the initial state. In our black holes case study, there is a specific range of temperatures such that black holes undergo a large-small-large phase transition by a monotonic changing of the pressure. This interesting phenomena has been first observed in a nicotine/water mixture Hudson (), and then seen in multicomponent fluids, binary gases, liquid crystals, and other diverse systems Narayanan (). In the context of black holes, the RPT has been observed for Born-Infeld adS black holes BIadS (), rotating adS black holes rotatingadS (); rotatingAmin (), de Sitter black holes deSitter (), hairy black holes hairy (), and adS black holes in massive gravity dRGTmassive (); massive (). The van der Waals like phase transition of SBH/LBH in dilaton gravity has been investigated for charged adS black holes Zhao (), and also, different types of nonlinear electrodynamics, such as power Maxwell invariant PMIdMo (); PMId (), exponential ENED (), and Born-Infeld PVdilatonH (); PVdilatonSh (). More recently a study regarding zeroth order SBH/LBH phase transition of charged dilaton black holes has been done Dehyadegari (). The purpose of this paper is studying the thermodynamics of -dimensional Born-Infeld-dilaton black holes and investigating the RPT in the canonical ensemble of extended phase space.

## Ii review of solutions and thermodynamics

Topological Born-Infeld-dilaton black holes in -dimensional spacetime were constructed by Sheykhi Sheykhi (). In what follows we concentrate our attention on the spherical symmetric black holes with negative cosmological constant. The line element reads

(3) |

where and are given by

(4) |

(5) |

in which is an arbitrary dilaton coupling constant determining the strength of coupling of the scalar and electromagnetic field, is an arbitrary constant, is nonlinearity (BI) parameter, and is a hypergeometric function. In addition, , , and is the dilaton field

(6) |

Using the definition of the surface gravity, one can obtain the Hawking temperature of the black hole on the outermost horizon, ,

(7) |

where . In addition, the entropy of the black hole per unit volume in Einstein gravity is a quarter of the event horizon area

(8) |

The electric potential , measured at infinity with respect to the horizon is given by the following explicit form

(9) |

where is an integration constant which is related to the electric charge of the black hole. One can use the flux of the electric field at infinity to obtain the electric charge per unit volume

(10) |

The total mass of obtained black holes per unit volume can be obtained by using the behavior of the metric at large

(11) |

It has been shown that by considering the entropy and electric charge as a complete set of extensive parameters, these conserved and thermodynamical quantities satisfy the first law of thermodynamics Sheykhi ()

(12) |

## Iii Reentrant Phase Transition

In the extended phase space, the negative cosmological constant is considered as a positive thermodynamical pressure Caldarelli (); Kastor (), which in our case it has the following form

(13) |

where in the absence of dilaton (), the pressure takes the standard form (1). In this situation, the total mass (11) plays the role of enthalpy of system Kastor (), and the Smarr formula and first law of thermodynamics are given by

(14) |

(15) |

where is the thermodynamical volume conjugate to

(16) |

Here, we study the thermodynamics of -dimensional Born-Infeld-dilaton black holes in the canonical ensemble (fixed and ) of extended phase space. So, by using the temperature (7) and the relation between the cosmological constant and pressure (13), it is straightforward to show that the equation of state, , is given by

(17) |

The thermodynamical behavior of the system is governed by the Gibbs free energy, therefore, we should obtain the Gibbs free energy as well. In the extended phase space, one can determine the Gibbs free energy per unit volume by using the following definition

(18) | |||||

where .

On the other hand, the heat capacity in extended phase space at constant pressure is

(19) |

which can be obtained easily by using Eqs. (7) and (8). It is worthwhile to mention that the heat capacity at constant is in fact a heat capacity at constant , , and because we are working in the canonical ensemble. The negativity of represents unstable black holes, whereas its positivity indicates stable ones. In order to study the phase transition of black holes, one can use the definition of inflection point

(20) |

which can be used to obtain the critical horizon radius and temperature . Due to complexity of obtained relation, we cannot calculate analytically. So, we use the numerical method and some digrams to investigate the RPT of the black holes for the fixed values of , , , and .

The general behavior of Born-Infeld-dilaton black holes is illustrated in Fig. 1. This figure have been plotted for different regions of temperature (pressure) in () diagram. These areas are equivalent to eachother. For in diagram, the curve looks like the Hawking-Page phase transition HP (). The dashed red line describes small unstable black holes with the negative heat capacity whereas the solid blue line corresponds to stable large black holes with the positive heat capacity (for more discussion about the relation between the heat capacity and Gibbs energy see MassiveYM ()). Considering Fig. 1, one can see that there is a critical point at in diagram (at in diagram which characterized by an inflection point). Black holes undergo a first order phase transition between small black holes (SBH) and large black holes (LBH) for and . For , there are three different phases of intermediate black holes (IBH), SBH, and LBH. The vertical line at shows a discontinuity in the Gibbs free energy which describes a zeroth order phase transition between SBH and IBH. There is also a first order SBH/LBH phase transition in this region of pressures and temperatures. This behavior is known as RPT. It is worthwhile to mention that IBH are macroscopically similar to LBH. Therefore, Black holes undergo the large-small-large phase transition in this region of pressures. Finally, there are just LBH for and .

Fig. 2 shows the coexistence lines of SBH+LBH (the dotted blue line) and IBH+SBH (the solid green line) in different scales. The dotted blue line is bounded by a critical point (,) and a triple point (,) between SBH, IBH, and LBH. Similarly, the solid green line is bounded by this triple point and another critical point (,). When black hole crosses the dotted (solid) line from left to right or top to bottom, it undergoes a first (zeroth) order phase transition from SBH to LBH (IBH to SBH). So, one can see the RPT behavior of Born-Infeld-dilaton black holes for a narrow range of temperatures and pressures .

## Iv Conclusions

In this paper, we have considered the cosmological constant as thermodynamical pressure and studied the thermodynamics of -dimensional Born-Infeld-dilaton black holes in the canonical ensemble of extended phase space. We have seen that in addition to the standard van der Waals like phase transition of these black holes PVdilatonH (); PVdilatonSh (), they can enjoy the RPT. It was shown that this behavior happens for a narrow range of temperatures and pressures. In this range of RPT, black holes undergo a zeroth order IBH/SBH phase transition and first order SBH/LBH phase transition.

###### Acknowledgements.

I wish to thank Shiraz University Research Council.## References

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