Phase transition and entropic force of de Sitter black hole in massive gravity
Yubo Ma, Yang Zhang, Lichun Zhang, Liang Wu, Ying Gao, Shuo Cao, Yu Pan
aa r X i v : . [ h e p - t h ] O c t Phase transition and entropic force of de Sitter black hole in massive gravity
Yubo Ma , Yang Zhang , Lichun Zhang , Liang Wu , Ying Gao , Shuo Cao , Yu Pan ∗ Institute of Theoretical Physics, Shanxi Datong University, Datong, 037009, China School of Mathematical and Statistical, Shanxi Datong University, Datong, 037009, China Department of Astronomy, Beijing Normal University, Beijing, 100875, China School of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China ∗ It is well known that de Sitter(dS) black holes generally have a black hole horizon and a cosmo-logical horizon, both of which have Hawking radiation. But the radiation temperature of the twohorizons is generally different, so dS black holes do not meet the requirements of thermal equilib-rium stability, which brings certain difficulties to the study of the thermodynamic characteristics ofblack holes. In this paper, dS black hole is regarded as a thermodynamic system, and the effectivethermodynamic quantities of the system are obtained. The influence of various state parameters onthe effective thermodynamic quantities in the massive gravity space-time is discussed. The condi-tion of the phase transition of the de Sitter black hole in massive gravity space-time is given. Weconsider that the total entropy of the dS black hole is the sum of the corresponding entropy of thetwo horizons plus an extra term from the correlation of the two horizons. By comparing the entropicforce of interaction between black hole horizon and the cosmological horizon with Lennard-Jonesforce between two particles, we find that the change rule of entropic force between the two system issurprisingly the same. The research will help us to explore the real reason of accelerating expansionof the universe.
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I. INTRODUCTION
In recent years, the study of thermodynamic properties of black holes has aroused great interest. Especially, aftertreating the cosmological constant as the thermodynamic pressure, the first law of black hole thermodynamics inextended phase space has been derived. Comparing the state parameters of black holes with that of the Van derWaals equation, the critical phenomena of various AdS space-time black holes have been studied. The influence ofvarious parameters on the phase transition is discussed. Some progress has been made in the study of black hole phasetransition [1–30]. For de Sitter space-time, space-time has black hole horizon and cosmological horizon. Both horizonshave Hawking radiation, and the radiation temperatures of the two horizons are generally different. Therefore dSblack holes do not meet the requirements of thermal stability, which brings certain difficulties to the study of thethermodynamic properties of black hole. In recent years, the study of the thermodynamic properties of de Sitterspace-time has attracted extensive attention [31–49]. Because in the early period of inflation, our universe is a quaside Sitter space-time, and the constant term introduced in the study of de Sitter space-time is the contribution ofvacuum energy, which is also a form of material energy. If the cosmological constant corresponds to dark energy,our universe will evolve into a new de Sitter phase. In order to construct the whole history of universe evolution, weshould have a clear understanding of the classical and quantum properties of the de Sitter space-time.Since the thermodynamic quantities corresponding to the two horizons in de Sitter space-time are all functions ofmass M , charge Q and cosmological constant Λ. There must be certain relation between the thermodynamic quantitiescorresponding to the two horizons. Considering whether the corresponding thermodynamic quantity in the de Sitterspace-time has the thermodynamic characteristics of AdS black hole after the correlation of black hole horizon anduniverse horizon; What is the relationship between the entropy of de Sitter space and the corresponding entropy oftwo horizons? What is the relationship between the entropic force caused by the interaction of two horizons and theratio of the position of the two horizons? These problems are very important to study the stability and evolutionof de Sitter space-time. Therefore, it is worth to establish a self-consistent relationship between the thermodynamicquantities of de Sitter space-time. In [50], the authors suggest a massive gravity which overcomes the traditionalproblems and yield an avenue for addressing the cosmological constant naturalness problem. Fierz and Pauli firstlyprovided the possibility of a massive graviton [51, 52]. Further, van Dam and Veltman [53] and Zakharov [54] hadfound the linear theory which was coupled to a source, they found that even when the mass of the graviton approaches ∗ e-mail:[email protected] to zero, the prediction of the theory is different from that of the linear theory. Later, Boulware-Deser ghost theorieshad been studied, which were nonlinear and ghostlike instability massive gravity theories [55, 56]. In the absence ofsuch instability, significant progress has been made in establishing the theory of massive gravity [57–62]. The moststraightforward way to construct the massive gravity theories is to simply add a mass term to the GR action, givingthe graviton a mass in such a way that GR is recovered as mass vanishes. Recently, a charged BTZ black holes hasbeen studied in [12, 63], which consider a massive gravity. The massive BTZ black holes in the presence of Maxwelland Born-Infeld electrodynamics in asymptotically (A)dS spacetimes was studied in [6]. In [64–67], thermodynamicsand entropy of AdS black hole in massive gravity were obtained, and the phase transition was discussed.In this paper, on the basis of effective thermodynamic quantity of de Sitter black hole in massive gravity (DSBHMG),the thermodynamic characteristics of de Sitter space-time are discussed. The condition of the phase transition ofDSBHMG space-time is found, and the influence of each parameter on the critical point is analyzed. It is notedthat DSBHMG space-time has a second-order phase transition similar to that AdS black hole when the parametersmeet certain conditions, except that the heat capacity at constant volume of DSBHMG is not zero. By calculatingthe entropic force of the interaction between the two horizons, the changing law of entropic force with the ratio ofthe position of two horizons is given. By comparing the curve with that of Lennard-Jones force [68] between twoparticles, it is found that the curve of entropic force changing with the position ratio of two horizons is similar to thecurve of Lennard-Jones force changing with the distance between two particles, which has the same change rule. Thisdiscovery provides a new way to study the internal cause of accelerating expansion of the universe.This paper is arranged as follows: for the continuity of this paper, in the second part, we briefly introduce thecorresponding thermodynamic quantities of black hole horizon and cosmological horizon in DSBHMG and the effectivethermodynamic quantities of DSBHMG. In the third part, we discuss the critical phenomena of DSBHMG. In thefourth part, we discuss the entropic force of interaction between the two horizons, and obtain the entropic force andthe Lennard-Jones force between two particles has a similar law of change. The fifth part of the paper discusses theconclusion. (We use the units G = h = k B = 1). II. THERMODYNAMICS OF BLACK HOLES IN MASSIVE GRAVITY
We consider (3 + 1) -dimensional massive gravity with a Maxwell field [65–67, 69, 70], the action is as follows S = 1 k Z d x √ g ( R − − F + m X i c i µ i ) , (1)where Λ is the cosmological constant and k = 0,1, or − F µν is the Maxwell field-strength tensor, c i are constants, and µ i are symmetricpolynomials of the eigenvalues of the matrix p g µν f µν where f µν is a fixed symmetric tensor. The action admits astatic black hole solution with the space-time metric and reference metric as ds = − f ( r ) dt + f − dr + r h ij dx i dx j , ( i, j = 1 ,
2) (2)where h ij dx i dx j is the line element for an Einstein space with constant curvature, f ( r ) is the metric function, whichis written by [71, 72]. f ( r ) = k − Λ3 r − m r + q r + c m r + m c , (3)The positions of black hole horizon and universe horizon satisfy the equation f ( r + ,c ) = 0, thus, the mass m canbe expressed in terms of r + ,c as M = m (cid:0) k + m c (cid:1) r c x (1 + x )2 (1 + x + x ) + q (1 + x ) (cid:0) x (cid:1) r c x (1 + x + x ) + r c m c x x + x ) , (4)where x = r + r c and Λ3 r c (cid:0) x + x (cid:1) = k − q r c x + c m r c (1 + x ) + m c , (5)The radiation temperature of black hole horizon and cosmological horizon is T + ,c = ± f ( r + ,c )4 π = 14 πr + ,c (cid:18) k − Λ r ,c − q r ,c + m c r + ,c + m c (cid:19) . (6)The thermodynamic quantities corresponding to the two horizons satisfy the first law of thermodynamics dM = T + ,c dS + ,c + V + ,c dP + µ + ,c dQ, (7)where V + ,c = v r ,c , P = − Λ8 π (8)when the radiation temperature T + of the black hole horizon is equal to that of the cosmological horizon T c , thecharge Q and Λ of the system satisfies the equation1 r + (cid:18) k − Λ r − q r + m c r + + m c (cid:19) = − r c (cid:18) k − Λ r c − q r c + m c r c + m c (cid:19) , (9)we can get the radiation temperature T when T + = T c . T = T + = T c = (1 − x )2 πr c (1 + x ) " k + m c r c (cid:0) x + x (cid:1) x ) + m c . (10)Recently, through the study of the thermodynamic characteristics of dS space-time, one can get the thermodynamicvolume of dS space-time is [33, 48] V = 4 π (cid:0) r c − r (cid:1) = 4 π r c (cid:0) − x (cid:1) , (11)if the energy M , charge Q and volume V in space-time are taken as the state parameters of a thermodynamic system,the first law of thermodynamics should be satisfied dM = T eff dS − P eff dV + φ eff dQ, (12)Considering the interaction between the black hole horizon and the cosmological horizon, we set the entropy of thesystem [47–49] S = πr c (cid:2) x + f ( x ) (cid:3) , (13)here the undefined function f ( x ) represents the extra contribution from the correlations of the two horizons. When theradiation temperature of black hole horizon and cosmological horizon is equal, the effective temperature of space-timeshould also be equal to the radiation temperature of two horizons T eff = T = T c = T + , (14)From equations (12), (13) and (14), the effective temperature T eff , pressure P eff and potential φ eff of the systemare obtained T eff = B ( x, q ) (cid:0) − x (cid:1) πr c x (1 + x ) , P eff = D ( x, q ) (cid:0) − x (cid:1) πr c x (1 + x ) , φ eff = (cid:18) ∂M∂Q (cid:19) S,V = 4 π Q (1 + x ) (cid:0) x (cid:1) v r c x (1 + x + x ) . (15)where B ( x, q ) = (cid:0) k + m c (cid:1) (cid:0) x − x + x + x (cid:1) (1 + x + x ) − q r c (cid:0) x + x − x + x + x + x (cid:1) x (1 + x + x ) + r c m c x (2 + x )(1 − x ) + 2 x x + x ) , (16) D ( x, q ) = (cid:0) k + m c (cid:1) (1 + x + x ) (cid:8) (1 + 2 x ) (cid:2) x + f ( x ) (cid:3) − x (1 + x ) (cid:0) x + x (cid:1) [2 x + f ′ ( x )] (cid:9) − q r c x (1 + x + x ) (cid:8) (cid:0) x + 3 x (cid:1) (cid:2) x + f ( x ) (cid:3) − x (1 + x ) (cid:0) x (cid:1) (cid:0) x + x (cid:1) [2 x + f ( x )] (cid:9) + r c m c x (1 + x + x ) (cid:8) x ) (cid:2) x + f ( x ) (cid:3) − x (cid:0) x + x (cid:1) [2 x + f ′ ( x )] (cid:9) . (17)Considering the initial conditions f (0) = 0, we can get f ( x ) = − (cid:0) − x − x (cid:1) − x ) + 85 (cid:0) − x (cid:1) / (18)we set the initial parameters to m = 2 . c = 2, c = 3 . q = 1 . r c = 1, k = 1, and then take different values for k , q , m , c and c respectively, to get the curve P eff − x and T eff − x curve (take r c = 1) as shown in FIG.1 and FIG.2 c = c = c = x P eff (a) c = c = c = x P eff (b) k =- (cid:0) = (cid:1) = x P eff (c) m = (cid:2) = (cid:3) = x (cid:7)(cid:8) P eff (d) q = (cid:9) = (cid:10) = x P eff (e) FIG. 1: P eff − x diagrams when the parameters change respectively. c = c = c = x T eff (a) c = c = c = x T eff (b) (cid:11) = (cid:12) = (cid:13) = (cid:14)(cid:15)(cid:16)(cid:17) x T eff (c) (cid:18) =- (cid:19) = (cid:20) = x (cid:21)(cid:22) T eff (d) (cid:23) = (cid:24) = (cid:25) = x T eff (e) FIG. 2: T eff − x diagrams when the parameters change respectively. III. CRITICAL PHENOMENA
Heat capacity of the system at constant volume: C V = T eff (cid:18) ∂S∂T eff (cid:19) V = T eff (cid:16) ∂S∂r c (cid:17) x (cid:0) ∂V∂x (cid:1) r c − (cid:0) ∂S∂x (cid:1) r c (cid:16) ∂V∂r c (cid:17) x (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂T eff ∂ rc (cid:17) x − (cid:16) ∂V∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c = 2 πr c B ( x, q ) x (cid:0) x (cid:1) (1 − x ) h B ′ ( x, q )(1 − x ) x ¯ B ( x, q ) − B ( x,q )(1+2 x +5 x − x ) x (1+ x ) i (19)where ¯ B ( x, q ) = (cid:0) k + m c (cid:1) (cid:0) x − x + x + x (cid:1) (1 + x + x ) − µ c (cid:0) x + x − x + x + x + x (cid:1) x (1 + x + x ) (20) B ′ ( x, q ) = dB ( x,q ) dx , and B ( x, q ) is presented by Eq.(16).From FIG.3, we know that the heat capacity C V of the system in DSBHMG is not zero, which is different from theresult for AdS black holes where C V = 0 .The heat capacity at constant pressure is C P eff = T eff (cid:18) ∂S∂T eff (cid:19) P eff = T eff (cid:16) ∂S∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c − (cid:0) ∂S∂x (cid:1) r c (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂T eff ∂r c (cid:17) x − (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c = 2 πr c B ( x, q ) E ( x, q ) F ( x, q ) (21)Here E ( x, q ) = (cid:2) x + f ( x ) (cid:3) " D ′ ( x, q ) (cid:0) − x (cid:1) − D ( x, q ) (cid:0) x + 5 x − x (cid:1) x (1 + x ) + [2 x + f ′ ( x )] ¯ D ( x, q ) (cid:0) − x (cid:1) x , (22) c = c = c = x - - C v (a) c = c = c = x - - C v (b) - =- . = / = x - - C v (c) = = = =>?@ x - - C v (d) A = B = C = x - - C v (e) FIG. 3: C V − x diagrams when the parameters change respectively and the initial parameters m = 2 . c = 2, c = 3 . q =1 . r c = 1, k = 1. ¯ D ( x, q ) = 2 (cid:0) k + m c (cid:1) (1 + x + x ) (cid:8) (1 + 2 x ) (cid:2) x + f ( x ) (cid:3) − x (1 + x ) (cid:0) x + x (cid:1) [2 x + f ′ ( x )] (cid:9) − µ c x (1 + x + x ) (cid:8) (cid:0) x + 3 x (cid:1) (cid:2) x + f ( x ) (cid:3) − x (1 + x ) (cid:0) x (cid:1) (cid:0) x + x (cid:1) [2 x + f ′ ( x )] (cid:9) + r c m c x (1 + x + x ) (cid:8) x ) (cid:2) x + f ( x ) (cid:3) − x (cid:0) x + x (cid:1) [2 x + f ′ ( x )] (cid:9) , (23) F ( x, q ) = ¯ B ( x, q ) " D ( x, q ) (cid:0) x + 5 x − x (cid:1) x (1 + x ) − D ′ ( x, q ) (cid:0) − x (cid:1) + ¯ D ( x, q ) " B ′ ( x, q ) (cid:0) − x (cid:1) − B ( x, q ) (cid:0) x + 5 x − x (cid:1) x (1 + x ) . (24) D ′ ( x, q ) = xD ( x,q ) dx , and D ( x, q ) is given by Eq.(17).From Eq.(21), the curves are depicted in FIG.4.The expansion coefficient is α = 1 V (cid:18) ∂V∂T eff (cid:19) P eff = 1 V (cid:16) ∂V∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c − (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x − (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c = 12 r c x (cid:0) x (cid:1) (1 − x ) F ( x, q ) " D ′ ( x, q ) (cid:0) − x (cid:1) − D ( x, q ) (cid:0) x + 5 x − x (cid:1) x (1 + x ) − ¯ D ( x, q ) x (25)The α − x curve is shown in FIG.5. c = c = c = DEF x - - C P eff (a) c = c = c = GHI x - - C P eff (b) J =- K = L = MNOP x - - C P eff (c) Q = R = S = TUVW
XYZ x - - C P eff (d) [ = \ = ] = ^_‘ x - - C P eff (e) FIG. 4: C P eff − x diagrams when the parameters change respectively and the initial parameters m = 2 . c = 2, c = 3 . q =1 . r c = 1, k = 1. c = c = c = abc x - - α (a) c = c = c = def x - - α (b) g =- h = i = jlno x - - α (c) p = r = s = tuvw xyz{ x - - α (d) | = } = ~ = (cid:127)(cid:128)(cid:129) x - - α (e) FIG. 5: α − x diagrams when the parameters change respectively and the initial parameters m = 2 . c = 2, c = 3 . q = 1 . r c = 1, k = 1. The isothermal compressibility is κ T eff = − V (cid:18) ∂V∂P eff (cid:19) T eff = 1 V (cid:16) ∂V∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c − (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂T eff ∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x − (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c = 48 πr c x (cid:0) x (cid:1) (1 − x ) F ( x, q ) " B ′ ( x, q ) (cid:0) − x (cid:1) − B ( x, q ) (cid:0) x + 5 x − x (cid:1) x (1 + x ) − ¯ B ( x, q ) x . (26)According to Eq.(26), we plot κ T eff − x curve in FIG.6. c = c = c = (cid:130)(cid:131)(cid:132) x - - κ T eff (a) c = c = c = (cid:133)(cid:134)(cid:135) x - - κ T eff (b) (cid:136) =- (cid:137) = (cid:138) = (cid:139)(cid:140)(cid:141)(cid:142) x - - κ T eff (c) (cid:143) = (cid:144) = (cid:145) = (cid:146)(cid:147)(cid:148)(cid:149) (cid:150)(cid:151)(cid:152)(cid:153) x - - κ T eff (d) (cid:154) = (cid:155) = (cid:156) = (cid:157)(cid:158)(cid:159) x - - κ T eff (e) FIG. 6: κ T eff − x diagrams when the parameters change respectively and the initial parameters m = 2 . c = 2, c = 3 . q =1 . r c = 1, k = 1. It can be seen from the curve C P eff − x , α − x , κ T eff − x , C P eff , α and κ T eff are divergence at x = x c . Theentropy and volume of the system are continuous. According to Ehrenfest’s classification of phase transition, x c isthe second-order phase transition point of the system.Gibbs function is given by G = M − T eff S = (cid:0) k + m c (cid:1) r c x (1 + x )2 (1 + x + x ) + q (1 + x ) (cid:0) x (cid:1) r c x (1 + x + x ) + r c m c x x + x ) − r c B ( x, q ) (cid:0) − x (cid:1) x (1 + x ) (cid:2) x + f ( x ) (cid:3) . (27)the G − T eff curve is followingFrom Figures C P eff − x , α − x , κ T eff and G − T eff , DSBHMG satisfies Ehrenfest’s second-order phase transitionconditions for phase transition classification at x = x c , so secondary phase transition occurs in DSBHMG space-timeat x = x c .From the expressions (11) and (13) of the total entropy and volume of space-time, it is known that in the range0 < x < G − T eff , DSBHMG doesnot have a first-order phase transition, which is different from the first-order phase transition of AdS black holes. c = c = c = (cid:160)¡ T eff - - - - - ¢ (a) c = c = c =
10 20 30 40 T eff - - - £ (b) ⁄ = ¥ = ƒ =-
10 20 §¤ T eff - - - - - - - ' (c) “ = « = ‹ = ›fifl(cid:176)
10 20 –† T eff - - - - - - ‡ (d) · = (cid:181) = ¶ =
10 20 •‚
40 50 60 T eff - - - - „ (e) FIG. 7: G − T eff diagrams when the parameters change respectively and the initial parameters m = 2 . c = 2, c = 3 . q =1 . r c = 1, k = 1. IV. THE ENTROPIC FORCE OF INTERACTION BETWEEN TWO HORIZONS
For a general thermodynamic system, when two systems are independent of each other, the corresponding entropyof system A and system B are S A and S B . When two systems have interactions, the entropy of the total system is S total = S A + S B + S AB (28)In the formula, the S AB is an extra entropy caused by the interaction of two systems. From formula (13), weknow that the total entropy of the effective thermodynamic system in DSBHMG is divided into two parts, one is theentropy corresponding to the two horizons, and the other term is the increase of the system entropy after consideringthe interaction of the two horizons as a thermodynamic system, so the entropy is S f = S AB = πr c f ( x ) = πr c " (cid:0) − x (cid:1) / − (cid:0) − x − x (cid:1) − x ) (29)Entropic force in thermodynamic system is [73–78] F = − T ∂S∂r (30)which T is the temperature of the system, r is the location of the boundary surface.The entropic force in DSBHMG is F = T eff (cid:18) ∂S f ∂r (cid:19) T eff (31)Where T eff is the equivalent temperature of the system, r = r c − r + = r c (1 − x ), from Eq.(29), we can obtainEq.(32),0 F ( x ) = T eff (cid:16) ∂S f ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c − (cid:16) ∂S f ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) (1 − x ) (cid:16) ∂T eff ∂x (cid:17) r c + r c (cid:16) ∂T eff ∂r c (cid:17) x = B ( x, q ) (cid:0) − x (cid:1) x (1 + x ) (cid:20) B ′ ( x, q ) (cid:0) − x (cid:1) f ( x ) − B ( x,q ) ( x +5 x − x ) x (1+ x ) f ( x ) + ¯ B ( x, q ) (cid:0) − x (cid:1) f ′ ( x ) (cid:21)h B ′ ( x, q ) (1 − x ) (1 − x ) − B ( x,q )(1+2 x +5 x − x ) x (1+ x ) (1 − x ) − ¯ B ( x, q ) (1 − x ) i . (32)The F ( x ) − x curve is shown in FIG.8. c = c = c = x - - ” (a) c = c = c = x - - » (b) … =- ‰ = (cid:190) = x - - ¿ (c) (cid:192) = ` = ´ = ˆ˜¯˘ x - - ˙ (d) ¨ = (cid:201) = ˚ = x - - ¸ (e) FIG. 8: F − x diagrams when the parameters change respectively and the initial parameters m = 2 . c = 2, c = 3 . q = 1 . r c = 1, k = 1. For different parameters, the intersection position of the F ( x ) − x curve and the axis is x . The entropic forcebetween the two horizons is positive( F >
0) when 1 > x > x , which means that the two horizons are mutuallyexclusive; and the entropic force between the two horizons is negative ( F <
0) when x > x >
0, indicating that thetwo horizons are mutually attractive. Therefore, the position ratio of the two horizons x is different, and the entropicforce between them is different. The two horizons are accelerated to separate under the action of entropic force, andthe cosmological horizon expands faster than the black hole horizon when 1 > x > x , and they are decelerating tobe separated by the entropic force, and the cosmological horizon undergoes a decelerating expansion relative to theblack hole horizon when x > x >
0. So we get that the different values S − s between the area S of the curveand the axis in interval 1 > x > x and the area S of the curve and the axis in interval x > x > S − S >
0. When it is equal to orless than zero, the cosmological horizon is changing from accelerating expansion to decelerating expansion, and thenchanging from accelerating contraction to decelerating contraction. One cycle ends and the next cycle begins. Thecosmological horizon is oscillating relative to the horizon of the black hole.Interactions between neutral molecules or atoms with a center of mass separation r are often approximated by theso-called Lennard-Jones potential energy φ L,J which is given by [68, 77–79] φ L,J = 4 φ min (cid:20)(cid:16) r r (cid:17) − (cid:16) r r (cid:17) (cid:21) , (33)where the first term is a short-range repulsive interaction and the second term is a longer-range attractive interaction.A plot of φ L,J φ min versus rr is shown in FIG.3. The value r = r corresponds to φ L,J = 0, and the minimum value of φ L,J is φ L,J φ min = − r min r = 2 ≈ .
122 (34)By the definition of potential energy, the force between a molecule and a neighbor in the radial direction from thefirst molecule is F r = − dφ L,J dr , which is positive (repulsive) for r < r min and negative (attractive) for r > r min . Whenthe center of the first particle coincides with the coordinate center, let the radius of the particle be r . When thecenter of the second particle is at r , and the boundary of the second particle is at r , then r = r + r . Let’s take y = r r , 0 < y ≤
1. Eq.(33) can be expressed as φ L,J ( y ) = 4 φ min (cid:20)(cid:16) r r (cid:17) − (cid:16) r r (cid:17) (cid:21) = 4 φ min " (cid:18) y y (cid:19) − (cid:18) y y (cid:19) . (35)The interaction between the two particles is F ( y ) = − dφ L,J dr = 4 φ min r (cid:20) (cid:16) r r (cid:17) − (cid:16) r r (cid:17) (cid:21) = 3 φ min r " (cid:18) y y (cid:19) − (cid:18) y y (cid:19) . (36) y - - - - ϕ L (cid:204) J ( ˝ ) FIG. 9: φ L,J ( y ) − y diagram when y from 0 to 1. y - - ˛ ( ˇ ) FIG. 10: F ( y ) − y diagram when y from 0 to 1. From FIG.8 and FIG.10, we can see the relation curve of entropic force between two horizons in DSBHMG with theratio of the position of the two horizons is very similar to that of the Lenard-Jones interaction force with the positionratio of the two particle boundaries, which has the same change rule.In order to show the similarity of the two curves more clearly, we show the two curves in the same coordinate as isshown in FIG.11.From FIG.11, we know that the trend of change in curve F ( x ) − x and F ( y ) − y is exactly the same, only theintersection position of the two curves and the abscissa is slightly different. It shows that the entropic force betweenthe horizon of black hole and the cosmological horizon is very similar to the Lenard Jones interaction force betweentwo particles in RN dssq space-time, which provides a new way to explore the interaction force between particles inblack hole. V. CONCLUSION AND DISCUSSION
When the black hole horizon and the cosmological horizon are viewed separately as independent thermodynamicsystems without considering the correlation between them, the space-time does not meet the requirements of thermo-dynamic equilibrium stability because the radiation temperature of the two horizons are different. Thus the space-timeis unstable. When the correlation of two horizons is considered, the effective temperature T eff , effective pressure P eff and effective potential φ eff reflecting the thermal properties of DSBHMG are given by Eq.(15). From curve C p eff − x , α − x and κ T eff − x , when the ratio of the positions of the two horizons in DSBHMG is x = x c , phase transitionoccurs in the system. According to Ehrenfest’s classification of phase transition, the phase transition of the system at2 — ( x )- x (cid:209) ( y )- y - - (a) FIG. 11: F ( x ) − x and F ( y ) − y diagrams when abscissa x and y from 0 to 1. this point is of second-order, which is similar to that of AdS black hole. In the study of the thermodynamic propertiesof spherically symmetric AdS black holes, people study the critical phenomena of AdS black holes by comparing thethermodynamic quantities of AdS black holes with the thermodynamic quantities of Van der Waals equation andobtain the critical exponents and the heat capacity of AdS black holes, which provides a basis for further study ofthe thermal effect of black holes and experimental observation. However, it is difficult to accept the fact that theheat capacity at constant volume of AdS black holes is zero. It is shown that the effective thermodynamic quantityof DSBHMG also has the phase transition characteristics similar to that of van der Waals system. From formula(19), the heat capacity of DSBHMG at constant volume is not zero, which is consistent with the result that this heatcapacity of van der Waals system is not zero. From the curve of C P eff − x , we know that the stability of DSBHMGdepends on the value of x . When x < x c , space-time meets the requirements of thermodynamic stability. When1 > x > x c , space-time does not meet the requirements of thermodynamic equilibrium stability, and space-timeis unstable. Therefore, there is no DSBHMG steady-state space-time satisfying the position ratio 1 > x > x c oftwo horizons in the universe, which provides a theoretical basis to find black holes. The influence of parameters inDSBHMG on space-time stability is shown in C P eff − x curve seeing FIG.4.In the framework of general relativity, the entropic force of interaction between the horizon of black hole and thecosmological horizon given by theory is very similar to the Lennard-Jones force between two particles confirmed byexperiments. Since the space-time metric is derived from relativity, the thermodynamics of space-time is based ongeneral relativity, and the thermal effect of black hole is obtained from quantum mechanics. The physical quantitiesobtained meet the first law of thermodynamics. Therefore, the conclusion we have given reveals the internal rela-tionship among general relativity, quantum mechanics and thermodynamics, and provide a new way for us to studythe interaction between particles in black holes and the micro state of particles in black holes, and the relationshipbetween Lennard-Jones potential and micro state of particles in ordinary thermodynamic systems.Through the analysis of the fourth part, it is known that under the action of entropic force, when the ratio of thepositions of the two horizons satisfies x < x <
1, the entropic force between the two horizons is mutually exclusive,and the two horizons in this region are accelerating expansion under the action of entropic force; when 0 < x < x ,the entropic force between the two horizons is mutually attractive, and the two horizons are decelerating expansion;until x →
0, the entropic force between the two horizons is zero, and the two horizons tend to be relatively static.Therefore, in the 0 < x <
Acknowledgments
We thank Prof. Z. H. Zhu for useful discussions.This work was supported by the Scientific and Technological Innovation Programs of Higher Education Institutionsof Shanxi Province, China (Grant No. 2020L0471, No. 2020L0472, No. 2019L0743) and the National Natural Science3Foundation of China (Grant Nos. 12075143, 11847123, 11475108, 11705106, 11705107, 11605107). [1] D. Kubiznak and R. B. Mann,
P-V criticality of charged AdS black holes, JHEP (2012) 033.[2] R. A. Hennigar, E. Tjoa and R. B. Mann, Thermodynamics of hairy black holes in Lovelock gravity, JHEP (2017) 70.[3] A. Rajagopal, D. Kubiznak and R. B. Mann, Van der Waals black hole, Phys. Lett. B (2014) 277.[4] S. H. Hendi, R. B. Mann, S. Panahiyan and B. Eslam Panah, van der Waals like behaviour of topological AdS black holesin massive gravity, Phys. Rev. D (2017) 021501(R).[5] S. H. Hendi, G. Q. Li, J. X. Mo, S. Panahiyan and B. Eslam Panah, New perspective for black hole thermodynamics inGauss-Bonnet-Born-Infeld massive gravity, Eur. Phys. J. C (2016) 571.[6] S. H. Hendi, B. Eslam Panah and S. Panahiyan, Einstein-Born-Infeld-massive gravity: AdS black hole solutions and theirthermodynamical properties, JHEP (2015) 157.[7] S. Panahiyan, S. H. hendi and N. Riazi, AdS dyonic black holes in gravity’s rainbow, Nucl. Phys. B (2019) 388.[8] R. G. Cai, L. M. Cao, L. Li and R. Q. Yang, P-V criticality in the extended phase space of Gauss-Bonnet black holes inAdS space, JHEP (2013) 005.[9] R. G. Cai, S. M. Ruan, S. J. Wang, R. Q. Yang and R. H. Peng, Complexity Growth for AdS Black Holes, JHEP (2016)161.[10] J. L. Zhang, R. G. Cai and H. W. Yu, Phase transition and Thermodynamical geometry of Reissner-Nordstr¨om-AdS BlackHoles in Extended Phase Space, Phys.Rev. D (2015) 044028.[11] J. L. Zhang, R. G. Cai and H. W. Yu, Phase transition and thermodynamical geometry for Schwarzschild AdS black holein
AdS × S spacetime, JHEP (2015) 143.[12] S. H. Hendi, B. Eslam Panah and S. Panahiyan, Massive charged BTZ black holes in asymptotically (a)dS spacetimes,JHEP (2016) 029.[13] W. Xu, H. Xu and L. Zhao, Gauss-Bonnet coupling constant as a free thermodynamical variable and the associated criti-cality, Eur. Phys. J. C (2014) 2970.[14] S. W. Wen and Y. X. Liu, Triple points and phase diagrams in the extended phase space of charged Gauss-Bonnet blackholes in AdS space, Phys. Rev. D (2014) 044057.[15] W. G. Brenna, R. B. Mann and M. Park, Mass and Thermodynamic Volume in Lifshitz Spacetimes, Phys. Rev. D (2015) 044015.[16] R. Banerjee, B. R. Majhi and S. Samanta, Thermogeometric phase transition in a unified framework, Phys. Lett. B (2017) 25.[17] R. Banerjee and D. Roychowdhury,
Thermodynamics of phase transition in higher dimensional AdS black holes, JHEP (2011) 004.[18] R. Banerjee and D. Roychowdhury, Critical behavior of Born Infeld AdS black holes in higher dimensions, Phys. Rev. D (2012) 104043.[19] M. S. Ma, R. Zhao and Y. S. Liu, Phase transition and thermodynamic stability of topological black holes in Ho ˇ r ava-Lifshitzgravity, Class. Quant. Grav. (2017) 165009.[20] M. S. Ma and R. H. Wang, Peculiar P-V criticality of topological Ho ˇ r ava-Lifshitz black holes, Phys. Rev. D (2017)024052.[21] S. H. Hendi, B. Eslam Panah, S. Panahiyan and M. S. Talezadeh, Geometrical thermodynamics and P-V criticality of theblack holes with power-law Maxwell field, Eur. Phys. J. C (2017) 133.[22] Z. Dayyani, A. Sheykhi and M. H. Dehghani, Counterterm method in Einstein dilaton gravity and the critical behavior ofdilaton black hole with a power-Maxwell field, Phys. Rev. D (2017) 84004.[23] D. C. Zou, Y. Q. Liu and R. H. Yue, Behavior of quasinormal modes and Van der Waals-like phase transition of chargedAdS black holes in massive gravity, Eur. Phys. J. C (2017) 365.[24] P. Cheng, S. W. Wei and Y. X. Liu, Critical phenomena in the extended phase space of Kerr-Newman-AdS black holes,Phys. Rev. D (2016) 024025.[25] M. Mir and R. B. Mann, Charged Rotating AdS Black Holes with Chern-Simons coupling, Phys. Rev. D (2017) 024005.[26] Z. X. Zhao and J. L. Jing, Ehrenfest scheme for complex thermodynamic systems in full phase space, JHEP (2014)037.[27] R. Zhao, H. H. Zhao, M. S. Ma and L. C. Zhang, On the critical phenomena and thermodynamics of charged topologicaldilaton AdS black holes, Eur. Phys. J. C (2013) 2645.[28] Zeinab, Dayyani and A. Sheykhi, Critical behavior of Lifshitz dilaton black holes, Phys. Rev. D (2018) 104026.[29] R. Zhou and S. W. Wei, Novel equal area law and analytical charge-electric potential criticality for charged Anti-de Sit-terblack holes, Phys. Lett. B (2019) 406.[30] J. M. Toledo and V. b. Bezerra,
Some remarks on the thermodynamic of charged AdS black holes with cloud of strings andquintessence, Eur. Phys. J. C (2019) 110.[31] S. Mbarek and R. B. Mann, Reverse Hawking-Page Phase Transition in de Sitter Black Holes, JHEP (2019) 103.[32] F. Simovic and R. B. Mann, Critical Phenomena of Charged de Sitter Black Holes in Cavities, Class. Quant. Grav. (2019) 014002.[33] B. P. Dolan, D. Kastor, D. Kubiznak, R. B. Mann and J. Traschen, Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes, Phys. Rev. D (2013) 104017.[34] S. H. Hendi, A. Dehghani and Mir Faizal, Black hole thermodynamics in Lovelock gravity’s rainbow with (A)dS asymptote,Nucl. Phys. B (2017) 117.[35] Y. Sekiwa,
Thermodynamics of de Sitter black holes: Thermal cosmological constant, Phys. Rev. D (2006) 084009.[36] D. Kubiznak, Thermodynamics of horizons: de Sitter black holes and reentrant phase transitions, Class. Quant. Grav. (2016) 24.[37] J. McInerney, G. Satishchandran and J. Traschen, Cosmography of KNdS Black Holes and Isentropic Phase Transitions,Class. Quant. Grav. (2016) 10.[38] M. Urano and A. Tomimatsu, Mechanical First Law of Black Hole Spacetimes with Cosmological Constant and Its Appli-cation to Schwarzschild-de Sitter Spacetime, Class. Quant. Grav. (2009) 105010.[39] S. Bhattacharya and A. Lahiri, Mass function and particle creation in Schwarzschild-de Sitter spacetime, Eur. Phys. J. C (2013) 2673.[40] R. G. Cai, Cardy-Verlinde formula and asymptotically de Sitter spaces, Phys. Lett. B (2002) 331 .[41] R. G. Cai,
Cardy-Verlinde formula and thermodynamics of black holes in de Sitter spaces, Nucl. Phys. B (2002) 375.[42] Y. B. Ma, S. X. Zhang, Y. Wu, L. Ma and S. Cao,
Thermodynamics of de Sitter Black Hole in Massive Gravity, Commun.Theor. Phys. (2018) 544-550.[43] M. Azreg-Aiou, Charged de Sitter-like black holes: quintessence-dependent enthalpy and new extreme solutions, Eur. Phys.J. C (2015) 34.[44] L. C. Zhang and R. Zhao, The critical phenomena of Schwarzschild-de Sitter Black hole, EPL (2016) 10008.[45] X. Y. Guo, H. F. Li, L. C. Zhang and R. Zhao,
Continuous phase transition and microstructure of charged AdS black holewith quintessence, Eur. Phys. J. C (2020) 168.[46] H. H. Zhao, L. C. Zhang, M. S. Ma and R. Zhao, P-V criticality of higher dimensional charged topological dilaton de Sitterblack holes, Phys. Rev. D (2014) 064018.[47] Y. B. Ma, Y. Zhang, L. C. Zhang, L. Wu, Y. M. Huang and Y. Pan, Thermodynamic properties of higher-dimensional dSblack holes in dRGT massive gravity, Eur. Phys. J. C (2020) 213.[48] L. C. Zhang, R. Zhao and M. S. Ma, Entropy of Reissner-Nordstr¨om de Sitter black hole, Phys. Lett. B (2016) 74.[49] Y. B. Ma, L. C. Zhang, T. Peng, Y. Pan and S. Cao,
Entropy of the electrically charged hairy black holes, Eur. Phys. J.C (2018) 763 .[50] K. Hinterbichler, Theoretical aspects of massive gravity, Rev. Mod. Phys. (2012) 671.[51] M. Fierz, On the relativistic theory of force-free particles with any spin, Helv. Phys. Acta (1939) 3.[52] M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field,Proc.Roy.Soc.Lond. A (1939) 211.[53] H. van Dam and M. J. G. Veltman,
Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B (1970)397.[54] V. I. Zakharov, Linearized gravitation theory and the graviton mass, JETP Lett. (1970) 312.[55] D. G. Boulware and S. Deser, Can gravitation have a finite range, Phys. Rev. D (1972) 3368.[56] D. G. Boulware and S. Deser, Inconsistency of finite range gravitation, Phys. Lett. B (1972) 227.[57] C. de Rham, G. Gabadadze, Generalization of the Fierz-Pauli action, Phys. Rev. D (2010) 044020.[58] C. de Rham, G. Gabadadze and A. J. Tolley, Resummation of Massive Gravity,Phys. Rev. Lett. (2011) 231101.[59] B. Eslam Panah and S. H. Hendi,
Black hole solutions correspondence between conformal and massive theories of gravity,EPL (2019) 60006.[60] B. Eslam Panah, S. H. Hendi and Y. C. Ong,
Black Hole Remnant in Massive Gravity, Phys. Dark Universe (2020)100452.[61] S. H. Hendi, G. H. Bordbar, B. Eslam Panah and S. Panahiyan, Neutron stars structure in the context of massive gravity,JCAP (2017) 004.[62] B. Eslam Panah and H. L. Liu, White dwarfs in de Rham-Gabadadze-Tolley like massive gravity, Phys. Rev. D (2019)104074.[63] S. H. Hendi, S. Panahiyan, S. Upadhyay and B. Eslam Panah, Charged BTZ black holes in the context of massive gravity’srainbow, Phys. Rev. D (2017) 084036.[64] B. R. Majhi and S. Samanta, P-V criticality of AdS black holes in a general framework, Phys. Lett. B (2017) 203.[65] R. G. Cai, Y. P. Hu, Q. Y. Pan and Y. L. Zhang,
Thermodynamics of Black Holes in Massive Gravity, Phys. Rev. D (2015) 024032 .[66] J. F. Xu, L. M. Cao and Y. P. Hu, P-V criticality in the extended phase space of black holes in massive gravity, Phys. Rev.D (2015) 124033.[67] S. Upadhyay, B. Pourhassan and H. Farahani, P-V criticality of first-order entropy corrected AdS black holes in massivegravity, Phys. Rev. D (2017) 106014.[68] D. C. Johnston, Thermodynamic Properties of the van der Waals Fluid, Online ISBN:
On Non-Linear Actions for Massive Gravity, JHEP (2011) 009.[70] A. Adams, D. A. Roberts and O. Saremi, Hawking-Page transition in holographic massive gravity, Phys. Rev. D (2015)046003.[71] D. C. Zou, R. H. Yue and M. Zhang, Reentrant pha AdS black holes in dRGT massive gravity, Eur. Phys. J. C (2017)256.[72] P. Boonserm and T. Ngampitipan, Greybody factor for black holes in dRGT massive gravity, Eur. Phys. J. C (2018)492. [73] E. P. Verlinde, On the origin of gravity and the laws of Newton, JHEP (2011) 029.[74] C. P. Panos and Ch. C. Moustakidis, A simple link of information entropy of quantum and classical systems with Newtonianr-2 dependence of Verlinde’s entropic force, Physica A (2019) 384.[75] D. E. Kharzeev,
Deconfinement as an entropic self-destruction: A solution for the quarkonium suppression puzzle, Phys.Rev. D (2014) 074007.[76] L. C. Zhang and R. Zhao, The entropic force in Reissner-Nordstr¨om-de Sitter spacetime, Phys. Lett. B (2019) 134798.[77] A. Plastino, M. C. Rocca and G. L. Ferri,
Quantum treatment of Verlinde entropic force conjecture, Physica A (2018)139.[78] A. Plastino and M. C. Rocca,
On the entropic derivation of the r-2 Newtonian gravity force, Physica A (2018) 190.[79] Y. G. Miao and Z. M. Xu,
On thermal molecular potential among micromolecules in charged AdS black holes, Phys. Rev.D98