Thermodynamics of the Reissner-Nordstr\"om-de Sitter Spacetime with Quintessence

For Anti-de Sitte (AdS) black holes, the isochoric heat capacity of system is vanished, while the isobaric heat capacity is not. However, this situation does not hold on for de Sitter (dS) black holes. In this work, by introducing the interaction between the black hole horizon and the cosmological horizon of the Reissner-Nordstr\"om-de Sitter (RNdS) spacetime with quintessence, we discuss the phase transition of this system. The results show that the spacetime not only has the similar phase transition behavior to that of Van der Waals (VdW) system, and the non-vanishing isochoric heat capacity fulfills the whole thermodynamics system. Through the discussion of the entropic force between two horizons, we find out the role of entropic force in the evolution of spacetime. In addition, we also study the influence of various parameters on the phase transition and entropic force, which will provide a new method for exploring the interaction among black hole molecules from a micro perspective.

a thermodynamic system. As early as 1970s, physicists such as Hawking and Bekenstein have established the four laws of black hole thermodynamics. [1][2][3][4] In order to explore the microstructure and evolution of black holes, people regard the cosmological constant of n-dimensional AdS spacetime as the state parameter of the black hole thermodynamic system -pressure P = n(n − 1) 16πl 2 , Λ = − n(n − 1) 2l 2 , (1.1) whose conjugate thermodynamic quantity is the volume Based on this issue, a series of studies on the thermodynamic behavior of AdS black holes have been done in the extended phase space [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Firstly, people found that the charged AdS black hole has a behavior similar to Van de Waals phase transition when choosing the quantities P and V as the independent dual variables. Subsequently, it has been proved in Ref. [24] that a charged AdS black hole possesses the Van der Waals (VdW) phase transition behavior by choosing Q 2 − Ψ as the independent dual variables. These developments of AdS black hole thermodynamic properties will not only help us understand the nature of black holes, but also help us to explore the microstructure of black holes [25][26][27][28][29][30][31][32][33][34].
In addition, since our universe during the inflationary period was a quasi-de Sitter spacetime, that drives our attention to a dS spacetime. Considering the cosmological constant in a dS spacetime as the dark energy, our accelerated expanding universe will evolve into a new de Sitter phase in the future [35]. Therefore, we should pay more attention on the relation between the classical, quantum, and thermodynamic properties of dS spacetime. While there are merely works on the thermodynamic behavior of black holes in a dS spacetime [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. The related thermodynamic properties of dS black holes will help us understand the relation between gravity and the conformal field theory. However, the investigation of the black hole thermodynamic properties in a dS spacetime is a complicated problem, since the absence of an everywhere timelike Killing vector outside the black hole horizon leads to an unclear definition of asymptotic mass. Furthermore, in a dS spacetime the black hole and cosmological horizons yield two distinct temperatures, which suggests that the system is in a non-equilibrium state.
The authors in Refs. [51,52] had shown that when the interaction between the black hole and cosmological horizons of the Reissner-Nordström-de Sitter black hole surrounded by quintessence (RN-dSQ) is considered, the entropy is the summation of the corresponding entropy for two horizons, and together with their interaction term, i.e.
where S c is the entropy of cosmological horizon and x is ratio of positions of two horizons and the interaction term reads The corresponding effective temperature satisfies T ef f = T + = T c , when the two horizon temperatures are equal. This result fulfills the requirements of ordinary thermodynamic systems. Therefore, the effective thermodynamic quantities of a dS spacetime are more general after considering the interaction between two horizons.
In Refs. [51,52] the authors assumed that there exists the interplay between the black hole and cosmological horizons in the RN-dSQ spacetime. And on this basis, they gave the effective thermodynamic quantities of spacetime, further discussed the thermodynamic properties of this system. Results shown that this system also has a continuous phase transition similar to the Van de Waals system or a charged AdS black hole, and parameters of this system influence the phase transition. However, a vanishing isochoric heat capacity in this case is unacceptable, which has aroused people's attention [53,54]. When the effective thermodynamic quantities are used to describe the thermodynamic properties of spacetime, the isochoric heat capacity will not vanish. It is shown that, under the consideration of interaction between two horizons, the method of the effective thermodynamic quantities to describe the thermodynamic properties of de Sitter spacetime has a more universal physical meaning. Furthermore, from the total entropy between two horizons in a RN-dSQ spacetime, the entropic force in the evolution of a dS spacetime plays a important role, which presents a new approach for studying the interaction of microscopic particles inside black holes and simulating the evolution of our accelerated universe.
This article is organized as follows. In Sec. II, we discuss the conditions for the existence of black hole and cosmological horizons in RN-dSQ spacetime, as well as the influence of spacetime parameters, and give the range of the position ratio x of two horizons. In Sec. III, we present the effective thermodynamic quantities of RN-dSQ spacetime, and analyse the relation between the effective temperature T ef f and the position ratio x of two horizons. The merely consistency between the effective temperature and black hole horizon temperature reminds us that when studying the spacetime thermodynamic properties of RN-dSQ, replacing the effective temperature T ef f with the black hole horizon radiation temperature T can simplify the problem. In Sec. IV, we extend the method of the phase transition for Van de Waals systems or charged AdS black holes, and use it to study the RN-dSQ spacetime, and obtain the phase transition properties of the RN-dSQ spacetime.
In Sec. V, we explore the entropic force between two horizons of the RN-dSQ spacetime, and obtain the function of entropic force with respect to position ratio x. It is found that the law of entropic force changing with x is very similar to the Lennard-Jones force between two microscopic particles. This phenomenon provides a new way for us to further study the interaction of black hole molecules. The Sec. VI is the summary. Kiselev considered the quintessence dark energy where ρ q = − 3αω 2r 3(ω+1) is the density of quintessence field, ω (−1 < ω < −1/3) is the quintessence dark energy barotropic index, α is the normalization factor associated with the quintessence density of dark energy and is always positive. On this issue, the static spherically symmetric solution of Einstein-Maxwell field equation for a Reissner-Nordström-dS (RN-dS) black hole in quintessence matters had been given by Refs. [55][56][57][58][59] with the metric function here M and Q represent the black hole mass and charge, l is the curvature radius of dS space. Now, let us analyze the existence of black hole and cosmological horizons, which are determined by the following equation Solving Eq. (2.5), one can obtain  given, we can obtain the relations between the minimum value of cosmological constant (l 2 min ) and horizon radius under which two horizons can both exist: For different values of Q, ω, α, by solving above equations we present the minimum values of the cosmological constant in Table. I.
In addition, from Eq. (2.5) we have where x ≡ r + /r c is ratio between the black hole horizon r + and cosmological horizon r c . At the points A and C, there exists the following equation  For simplification, we choose 3α r 1+3ω where β + is the quintessence field on the black hole horizon. Then Eq. (2.9) can be rewritten By combining Eq. (2.8) and Eq. (2.9), r + can be solved (2.14) Using the definition x min = r A /r E shown in Fig. 1, we can plot the x min − l 2 curves with respect to variant parameters by substituting Eq. (2.12) into Eq. (2.11). And the numerical results of x min for different parameters are shown in Table. II. The results show that the minimum value x min is merely independent of β + and l 2 , but it is more sensitive to ω, conversely.
According to Eq. (2.11), one has The above equation implies that the minimum radius r A and the maximum radius r C of black hole horizon position in RN-dSQ spacetime are independent of the value of ω. Taking Q=1, β + = 0.0001, 0.0005, 0.0008, one can draw the curves of r A,C − l 2 , where one value of l 2  correspond to two values of r, i.e. r A and r C . Therefore, the values of M A and M C depend on l 2 once β + and Q are fixed. Fig. 4 shows that r C increases with the increase of l 2 , while r A remains nearly unchanged, which is consistent with the fact that x min is independent on To study the thermodynamic properties of RN-dSQ spacetime , we need to study the thermodynamic properties of RN-dSQ spacetime endowed with the black hole horizon and the cosmological horizon. While the existence of these two horizons is determined by their position ratio x. Fig. 2(a) implies that the range of x is x min ≤ x ≤ 1 once the spacetime parameters are set, and x min is affected by parameters as shown in Figs. 3(a) and 3(b).
Therefore, we adopt x min ≤ x ≤ 1 as the range of ratio x to study the thermodynamic properties of RN-dSQ spacetime .

III. EFFECTIVE THERMODYNAMIC QUANTITIES OF RN-DSQ SPACETIME
For the RN-dSQ spacetime, we focus on the space between the black hole and the cosmological horizon. Thus the volume of system reads [40,41,49,50] When introducing the interaction between the two horizons, the total entropy of system is the sum of two individual horizon entropies and their interactive term, which is a function of positions of the two horizons Here the undefined function f (x) represents the extra contribution from the correlations of two horizons. Based on the first law of black hole thermodynamics [49][50][51][52]: and Eqs. ) ) For the given effective temperature T ef f , from Eq. (3.5) the inverse of black hole horizon radius satisfies 1 Once P ef f is given, r 2 + satisfies the following equation according to Eq. (3.5) In order to further understand the effective temperature of this system, we should compare it with the temperatures on two horizons. From Eq. (2.5) we can obtain the Hawking temperatures of the black hole and the cosmological horizons as follows: . This implies the effective temperature can be replaced with the one on black hole horizon.
In addition, the influence of parameters on the effective temperature is exhibited in Fig.   6, from which we can see that the behavior of temperature as a function of x is merely  independent of ω and β + when l 2 is given. However, for the fixed parameters Q ,ω, and β + , the behavior of T ef f with respect to x is only affected by l 2 .

IV. PHASE TRANSITION OF RN-DSQ SPACETIME
In this section, we would like to study the phase transition in the canonical ensemble.
The Gibbs free energy of RN-dSQ spacetime reads The critical point denotes a second-order phase transition which is determined by the fol- According to Eq.  Fig. 7. It shows that the value of ω has no effect on the isothermal curve of P ef f − V , while the curve of P ef f − V is more sensitive to the parameter β + .
It is shown in Fig. 7(a) that the isotherm of equation of state is in the range of P min ef f < P ef f < P max ef f , there are three possible values of V which correspond to the same value of P ef f . Furthermore, for the range of P min ef f < P ef f < P max ef f , the slope of curve is positive, i.e.,   The isobaric heat capacity of spacetime is  Fig. 9(a). The volume expansion coefficient reads Then substituting Eq. (3.8) into Eqs. (3.5) and (4.4), we plot the isobaric curve of β − T ef f in Fig. 9(b). The isothermal compression coefficient becomes Substituting Eq. (3.7) into Eqs. (3.6) and (4.5), we plot the isothermal curve of κ T ef f −P ef f in Fig. 9(c). Fig. 9 shows that RN-dSQ spacetime presents typical continuous phase transition characteristics, i.e., it possesses the phase transition characteristics of VdW system or AdS black holes. These curves also reflect the influence of parameter β + on the critical point.
The isochoric heat capacity of spacetime is  Fig. 10. It can be seen from Fig. 10 that, the isochoric heat capacity of spacetime doesn't vanish. That is different with AdS black holes, whose isochoric heat capacities are zero [25][26][27][28]. For any thermodynamic systems, the isochoric heat capacity doesn't vanish in general, so our conclusion is more universal. Therefore, when considering the interaction of two horizons for the RN-dSQ spacetime, the effective thermodynamic quantities and corresponding equations are more closer to the characteristics of an ordinary thermodynamic system. Thus this thermodynamic system described by the effective thermodynamic quantities is more universal.
According to Ehrenfest's classification of phase transitions, if the Gibbs functions of two phases and their first-order partial derivatives are continuous, but their second-order partial derivatives are of sudden changes, the phase transition is called the second-order one. From the above analysis, we know that for the lower effective temperature (T ef f < T c ef f ), the entropy and volume will undergo sudden changes, and the spacetime undergoes a first-order phase transition. As T ef f = T c ef f , the entropy and volume are continuous, while the isobaric heat capacity, isobaric expansion coefficient, and isothermal compression coefficient both have a peak at the critical point shown in Fig. 9.

V. ENTROPIC FORCE BETWEEN TWO HORIZONS
The recent exploration of the interactive force between microscopic particles inside black holes has attracted widespread attention [25,[62][63][64][65]. In order to understand the microstructure of RN-dSQ spacetime more clearly, we will check out the behavior of entropic force between two horizons in this section. The entropic force in a thermodynamic system can be defined as follows [50,66,67] where T is the temperature and r is the radius of system. In this part we pay attention on the entropic force between the black hole and cosmological horizons. According to Eq. where T ef f stands for the effective temperature of this system and r ≡ r c − r + . From the above equation, we havẽ Then substituting Eq. (3.5) into Eqs. (4.2) and (5.4), we can get theF (x) − x curves as shown in Fig. 11.
It is obviously that the entropic force between two horizons depends on the cosmological constant l 2 , while merely independent of other parameters β + and ω. For the far distance between two horizons (i.e., the small value of x), the entropy force is negative, which indicates there exists an attractive force between two horizons to make them more closer. Up to the certain distance (i.e., the certain value of x), the entropic force is zero. Continuously decreasing the distance, the interactive entropy force between two horizons becomes positive, and it will approach to infinity when two horizons coincide. In the case of x = 1, the infinite entropic force prompts them to accelerated separate.

VI. CONCLUSION AND DISCUSSION
In this work, we introduced the interaction between the black hole and cosmological horizons and presented the effective thermodynamic quantities of the RN-dSQ spacetime.
The related thermodynamical properties of system were also investigated. Through the analysis in Section IV, we know that the RN-dSQ spacetime also have the continuous phase transitions and first-order ones, that is similar to VdW system or AdS black holes. The results showed that the effective temperature and phase transition point are independent of the RN-dSQ spacetime quintessence dark energy barotropic index ω. The existence of the black hole and cosmological horizons are determined by the cosmological constant and the normalization factor associated with the quintessence density of dark energy α when other parameters are given. Based on the effective quantities of system, we found that the isochoric heat capacity is non-zero, and it increases with the effective temperature and quintessence field β + . This conclusion is similar to ordinary thermodynamic systems, so our results have more general physical meanings.
From the curve ofF (x) − x, the characteristic of the change of entropic force with x is: when x → 1, the entropic force tends to infinity, which indicates that the infinite entropic force leads to accelerated separation between two horizons. At the same time, two horizons will be accelerating separattion under the infinity entropic force, and the energy M of system will decrease. Continuously decreasing the value of x, the entropic force will become negative and prevent the separation of two horizons. On the other hand, from the curve T ef f − x we found that for the low effective temperature (i.e., the large value of x), the entropic force between the two horizons promotes the separation of two horizons. While for the high effective temperature (i.e., the small value of x), the entropic force prevents the separation of the two horizons. These conclusions will provide a theoretical foundation for studying the evolution of spacetime.
In addition, comparing the entropic force between two horizons with the Lennard-Jones force between two particles, we found that the curves obtained by completely different methods for different systems are so surprisingly similar. The Lennard-Jones force of two particles is inherently related to the entropic force between two horizons. To further study the thermodynamic properties of black holes, it is necessary to explore the microstructure of system.
We had shown the interactive force between two horizons, which reflected the interactive force between microscopic particles on the curved surfaces (the black hole and cosmological horizons) in dS spacetime. The method can be extended to study the interaction of microscopic particles inside black holes, which will provides a new way to explore the interaction between black hole molecules.