Thermodynamic Phase Transition And Geometry For Quintessence Black Holes

Thermodynamics of black holes continues to be an important and fascinating topic in gravitational physics.In particular,motivated by the AdS/CFT correspondence,ther-modynamics for anti-de Sitter(AdS)black holes has received great attention.Hawking and Page found that,in an AdS space,there exists a phase transition between the stable large Schwarzschild black hole and the thermal gas.In the spirit of the correspondence,this phase transition is interpreted as the confinement/deconfinement phase transition in the dual strongly coupled gauge theory.More interestingly,it was also found that there is a small-large black hole phase transition for charged or rotating AdS black holes,which enlarges our knowledge on the black hole thermodynamics.In recent years,in AdS space-time,a focus of black hole thermodynamics is to consider the cosmological constant as the thermodynamic pressure of the thermodynamic system,and the black hole mass as the enthalpy.Adopting this interpretation of the cosmological constant,the thermodynamics law for the AdS black hole is completely established,just like the law of general thermo-dynamics system.More and more studies imply that the black hole system has rich phase structure.Here,we will also use this interpretation of cosmological constants to study the thermodynamic properties of quintessence black holes and their corresponding phase transition properties.First,we will study the effects of quintessence dark energy on the thermodynamic properties and phase transitions of black holes in the extended phase space of the black hole.The oscillation behavior of isotherm on the pressure-volume plane can reveal and identify the relevant properties of the black hole phase transition and critical situation.Based on the understanding of the thermodynamic quantities of quintessence black holes,we will study the oscillation for the isotherm.And with the help of isotherms,the critical point can be determined by the property that the oscillation just disappears at the critical point of the phase transition.Further,based on the free energy swallow tail behavior,we will extend the Maxwell equal area law and Chapeyron equation,and lay the foundation for further studying the properties for the first-order phase transition of small-large black holes.Then starting from the equation of state of the black hole,we focus on the critical exponent and scaling law,and discuss the relationship between the black hole phase transition and that described by the mean field theory under the background of the quintessence dark energy.Secondly,it is an important means to study the thermodynamic phase transition from thermodynamic geometry to the study of phase transition.Therefore,based on the Hessian matrix,we construct three kinds of thermodynamic geometries of quintessence black holes in the parameter space,i.e.,the Weinhold geometry,the Ruppeiner geometry,and the free energy geometry.For these three different geometries,we calculate their geometric curvature respectively.Further,based on the specific form of scalar curvature,the divergence behavior is discussed,and is compared with the point of transition and critical point of black hole.These reveal the relationship between the black hole transition point and the singular behavior of the curvature.The study of thermodynamic phase transition and thermodynamics of quintessence black hole will help us understand the influence of quintessence dark energy on the thermo-dynamics and thermodynamic geometry of black holes,and will enrich our understanding of thermodynamics of black holes.