Entropic force between two horizons of dilaton black holes with a power-Maxwell field
aa r X i v : . [ h e p - t h ] J a n Entropic force between two horizons of dilaton black holes with a power-Maxwell field
Hui-Hua Zhao , Li-Chun Zhang ∗ , Ying Gao , Fang Liu Institute of Theoretical Physics, Shanxi Datong University, Datong, 037009, China School of Mathematics and Statistics, Shanxi Datong University, Datong, 037009, China
In this paper, we consider ( n +1)-dimensional topological dilaton de Sitter black holes with power-Maxwell field as thermodynamic systems. The thermodynamic quantities corresponding to the blackhole horizon and the cosmological horizon respectively are interrelated. So the total entropy of thespace-time should be the sum of the entropies of the black hole horizon and the cosmological horizonplus a corrected term which is produced by the association of the two horizons. We analyze theentropic force produced by the corrected term at given temperatures, which is affected by parametersand dimensions of the space-time. It is shown that the change of entropic force with the positionratio of two horizons in some region is similar to that of Lennard-Jones force with the position ofparticles. If the effect of entropic force is similar to that of Lennard-Jones force, and other forcesare absent, the motion of the cosmological horizon relative to the black hole horizon would have anoscillating process. The entropic force between the two horizons is probably one of the participantsto drive the evolution of universe. Keywords : entropy, entropic force, dilaton dS space-time
PACS numbers: 04.70.-s, 05.70.Ce
I. INTRODUCTION
In the early cosmic inflation, our universe is a quasi-asymptotic de Sitter (dS) space-time, where the introducedcosmological constant term can be seen as the vacuum energy. If the cosmological constant corresponds to darkenergy, our universe will evolve into a new de Sitter phase. In order to construct the whole evolutionary history forour universe and find out the reason of the accelerated expansion as well, the classical, quantum and thermodynamicproperties of dS space-time should be studied. In addition, the success of the correspondence between Anti-de Sitterspace and conformal field theory (AdS/CFT) has prompted people to find a similar dual relationship for de Sitterspace-time.In recent years, the thermodynamic properties and possible phase transition of dS black holes have been studiedextensively [1–22]. For a dS black hole there are a black hole horizon and a cosmological horizon, and, in general,the radiation temperatures of the two horizons are different. Considering the two horizons as two thermodynamicsystems they satisfy the first law of black hole thermodynamics respectively, but their thermodynamic quantities areinterrelated because of their common quantities, mass, electric charge, and cosmological constant. In most of theprevious works on thermodynamic properties of dS black holes, the whole entropy of dS black hole is seen as the sumof the entropies of black hole horizon and cosmological horizon. Considering the correlation of the two horizons, acorrected term for the whole entropy is required, which is derived and analyzed in this work.In 2011, Verlinde [23] thought of linking gravity to an entropic force. Gravity emerges as a consequence of informa-tion regarding the positions of material bodies, combining a thermal gravitation treatment to ’t Hooft’s holographicprinciple. The ensuing conjecture was later proved [24–31] in a classical scenario. Accordingly, gravitation ought to beviewed as an emergent phenomenon. Such exciting Verlinde’s idea received a lot of attention[32–37]. So the entropicforce is an important force in the universe and it is probably one of the participants to drive the cosmic acceleratedexpansion.As an explanation for the cosmic accelerated expansion, the early theory of dark energy has been proposed by Riess[38–41]. In this theory, the cosmic accelerated expansion is caused by an exotic component called dark energy, whichaccounts for about 73% of the universe’s capacity according to astronomical observations. Astronomers assume thatdark energy exists in the first second after the Big Bang. the Big Bang pushes all matter to the whole space, andthen the initial expansion begins. Shortly after the Big Bang, dark energy bumps several times, which causes thepresent cosmic accelerated expansion. Some models for dark energy evolution have been proposed. For example, ifthe equation of state of dark energy is P = ωρ , where ω is the parameter of the equation of state, the evolution ofdark energy is exponential with the power 3(1 + ω ) when ω > − ∗ e-mail:[email protected](L.-C. Zhang), corresponding author In addition, the possible reason for the cosmic accelerated expansion is dark matter, which interacts more stronglywith normal matter or radiation than previously assumed. The existence of dark matter in the universe has beena common sense of modern cosmology. Dark matter accounts for about 23% in the total cosmic components. Darkmatter does not participate in electromagnetic interaction, nor interact with photons. The latest research shows[47–49] that dark matter appeared earlier than normal matter during the expansion of the universe, although normalmatter is produced during the Big Bang. A kind of non-spin scalar particle is produced during the rapid cosmicexpansion. Up to now, only one type of scalar particle has been found, which is the famous Higgs boson. Accordingto Tenkanen, it can be the candidate of dark matter. But how do the dark energy, the dark matter and the totalenergy of the universe evolve in the expansion universe? The real reason for the cosmic accelerated expansion are stillnot clear.The fact that the Universe expands with acceleration along the scheme of standard Friedmann model[40] createdmuch more interest in the alternative theories of gravity in recent years, one of which is dilaton gravity. Dilatongravity can be thought as the low energy limit of string theory, and one recovers Einstein gravity along with ascalar dilaton field, which is nonminimally coupled to the gravity and other fields such as gauge fields. We areinterested in studying the properties of the dilaton black holes when the gauge field is in the form of the power-Maxwell field[50]. Being different from the linear electromagnetic field, nonlinear electrodynamics was introduced toremove the central singularity of the point-like charges and obtained finite energy solutions for particles by extendingMaxwell’s theory. In cosmology, one can call upon nonlinear electrodynamics to explain the inflationary epoch andthe late-time accelerated expansion of the universe[51, 52]. A variety of nonlinear electrodynamics models[53–58]have been proposed and studied extensively, in which the power-Maxwell field[54] is conformally invariant in ( n + 1)-dimensional space-time for p = ( n + 1) /
4, where p is the power parameter of the Power-Maxwell Lagrangian, whilethe Maxwell Lagrangian is only conformally invariant in four dimensions.The effect of the dilaton field [15, 59, 60]and the power-law Maxwell field[61–64] on thermodynamics of anti-deSitter (AdS) black holes has been studied in extended phase space. However, as far as we know, the discussion forthe effective thermodynamic quantities with effects of the power-law Maxwell field and the dilaton field has not beendone in de Sitter space-time. In this paper, we study entropy and entropic force of dilaton black holes coupled tononlinear power-Maxwell field in de Sitter space-time, and investigate the effects of exponent p , the dilaton couplingconstant α and the space-time dimension n on the entropy and the entropic force of the black holes in dS space-timeand to explore their connection to the expansion of the universe.This paper is organized as follows. The solutions of charged dilaton black hole with power-Maxwell field in dS space-time are introduced in section 2. The thermodynamic quantities respectively corresponding to the black hole eventhorizon and the cosmological envent horizon in the dS space-time are given in section 3. Considering the correlationof two horizons the effective thermodynamic quantities and the modified entropy are derived in section 4. In section5, according to the relationship between entropy and entropic force, the entropic force between the two horizons isobtained and analyzed. Conclusions and discussions are given in the last section. The units G n +1 = ~ = k B = c = 1will be used throughout this work. II. BLACK HOLE SOLUTION OF EINSTEIN POWER-MAXWELL-DILATON FIELDS IN DSSPACE-TIME
In this section, we introduce the action of Einstein power-Maxwell-dilaton (EPMD) gravity, the solutions of theEPMD field equations, and the mass, the electric charge, and the cosmological constant of the EPMD black holes indS space-time.The action of ( n + 1)-dimensional ( n ≥
3) EPMD gravity can be written as [50, 59, 65–67] I = − π Z d n +1 x √− g (cid:20) R − n − ∇ Φ) − V (Φ) + ( − e − α Φ / ( n − F µν F µν ) p (cid:21) , (1)which can yield the follow field equations by taking the action as varying with respect to the gravitational field g µ,ν ,dilaton field φ and the gauge field F µ,ν , R µν = (cid:20) n − V (Φ) + 2 p − n − (cid:16) − F e − α Φ / ( n − (cid:17) p (cid:21) g µν + 4 n − ∂ µ Φ ∂ ν Φ + 2 pe − αp Φ / ( n − ( − F ) p − F µλ F ν λ , (2) ∇ Φ − n − ∂V∂ Φ − αp e − α Φ / ( n − ( − F ) p = 0 , ∂ ν h √− ge − α Φ / ( n − ( − F ) p − F µν i = 0 , (3)where R is the Ricci scalar, V (Φ) is a potential for Φ, p and α are two constants determining the nonlinearity ofthe electromagnetic field and the strength of coupling of the scalar and electromagnetic field, respectively. F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor and A µ is the electromagnetic potential.The topological black hole solutions take the form [50, 59, 65–67] ds = − f ( r ) dt + dr f ( r ) + r R ( r ) d Ω n − , (4)where f ( r ) = − Ar γ − mr ( n − − γ ) − + q p Br − n − p +1] − p ( n − γ p − + C Λ r − γ ) , (5)and A = k ( n − α +1) b − γ ( α − α + n − , B = p p ( α +1) (2 p − b − n − pγ (2 p − α + n − p ) , C = α +1) b γ ( n − α − n ) , in which b is an arbitrary nonzeropositive constant, γ = α / ( α + 1) , Π = α + ( n − − α ) p .Note that Λ remains as a free parameter and Λ > n − α )( n − l , (6)where l denotes the ADS length scale. In the Eq.(5), m appears as an integration constant and is related to theADM (Arnowitt-Deser-Misnsr) mass of the black hole. According to the definition of mass due to Abbott and Deser[68, 69], the mass of the solution Eq.(5) is [70] M = b ( n − γ ( n − π ( α + 1) m, (7)The electric charge Q and potential U are expressed as[50] Q = 2 p − q p − π , U = ( n − p qb (2 p − n +1) γ p − ΠΥ r Υ , Υ = n − p + α (2 p − α ) . (8)The fact that the electric potential U should have a finite value at infinity and the term including m in the solution f ( r ) in spacial infinity should vanish lead to the restrictions on p and α [61].12 < p < n + α . (9) α < n − . (10) III. THERMODYNAMIC QUANTITIES OF THE TWO HORIZONS OF THE EPMD BLACK HOLESIN DS SPACE-TIME
The dS black holes have two horizons, which are BEH and CEH. The BEH locates at r = r + and the CEH locates at r = r c . The positions of them can be determined by f ( r + ) = 0 and f ( r c ) = 0 respectively. Thermodynamic quantitieson BEH and CEH satisfy the first law of thermodynamics respectively [3, 8, 13]. In this section, we introduce thethermodynamic quantities corresponding to the BEH and CEH respectively. Replace the r in the Eq.(8) with r + or r c , one can get the electric potentials of BEH or CEH.The surface gravities of BEH and CEH are respectively given by κ + = 12 df ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) r = r + = (1 + α )2 k ( n − b − γ (1 − α ) r γ − − b γ n − r − γ + − p p (2 p − b − n − γp p − q p Π r p ( n − − γ )+12 p − , (11) κ c = − df ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) r = r c = − (1 + α )2 k ( n − b − γ (1 − α ) r γ − c − b γ n − r − γc − p p (2 p − b − n − γp p − q p Π r p ( n − − γ )+12 p − c , (12)from which the radiation temperatures of the two horizons can be got by T + ,c = κ + ,c / π .When f ( r + /c ) = 0 , we have m ( r + ) = − Ar α + n − α +1+ + Bq p r − α − p + n (2 p − α +1)+ + C Λ r − α − nα +1+ ,m ( r c ) = − Ar α + n − α +1 c + Bq p r − α − p + n (2 p − α +1) c + C Λ r − α − nα +1 c . (13) m ( r + ) = m ( r c ) = m , and take x = r + /r c as the ratio of positions of BEH and CEH, which meets 0 < x ≤
1, then C Λ =
A r α − α +1 c (1 − x α + n − α +1 )(1 − x − α − nα +1 ) − Bq p r p (1+ α − n ) − α (2 p − α +1) c (1 − x − α − p + n (2 p − α +1) )(1 − x − α − nα +1 ) , (14) m = Ar α + n − α +1 c x − α − nα +1 − x α + n − α +1 (1 − x − α − nα +1 ) + Bq p r − α − p + n (2 p − α +1) c x − α − p + n (2 p − α +1) − x − α − nα +1 (1 − x − α − nα +1 ) . (15)The thermodynamic volumes corresponding to the two horizons are respectively given by V + = ( α + 1) b γ ( n +1) ω n − n − α r n − α α +1+ ,V c = ( α + 1) b γ ( n +1) ω n − n − α r n − α α +1 c , (16)where ω n − represents the volume of constant curvature hypersurface described by d Ω k,n − .The entropies of BEH and CEH in dS space are expressed respectively as S + = b ( n − γ r ( n − − γ )+ ,S c = b ( n − γ r ( n − − γ ) c . (17) IV. EFFECTIVE THERMODYNAMICS AND MODIFIED ENTROPY OF THE EPMD BLACK HOLESIN DS SPACE-TIME
In general, the radiation temperatures of the BEH and CEH are different. So, If one investigates the black hole in dSspace-time including BEH and CEH as a whole thermodynamic system, it is usually thermodynamically unstable ornon-equilibrium. We find that the radiation temperatures are equal if the charge of the system satisfies some condition.Under the condition, considering the correlation of the two horizons, we derived the effective thermodynamic quantitiesand the modified entropies of the EPMD black holes in dS space-time.When the radiation temperatures of BEH and CEH are equivalent, κ + = κ c , Eq.(11) and Eq.(12) with Eq.(14) givethe condition about the charge for the same radiation temperature of BEH and CEH.2 p pq p (2 p − b − n − pγ +2 γ (2 p − r p (3 − n − α ) − p − α +1) c Π = kA ( x ) B ( x ) ( n − α − , (18)where A ( x ) = ( α − n )( α + n −
2) (1 − x α + n − α +1 )(1 + x − α α ) + (1 + x α − α +1 )(1 − x − α − nα +1 ) ,B ( x ) = ( α − n )(2 p − α + n − p ) (1 + x − α α )(1 − x − α − p + n (2 p − α +1) ) − (1 + x − p ( n − − γ )+12 p − )(1 − x − α − nα +1 ) . (19)Substituting Eq.(14) and Eq.(18) into Eq.(12), the temperature T for the same radiation temperature of BEH andCEH can be obtained as T = T + = T c = − (1 + α )4 π kr ( α − α +1 c b − γ ( n − − α ) α − n )( α + n −
2) (1 − x α + n − α +1 )(1 − x − α − nα +1 ) − A ( x ) B ( x ) − ( α − n )(2 p − α + n − p ) (1 − x − α − p + n (2 p − α +1) )(1 − x − α − nα +1 ) . (20)Substituting Eq.(18) into Eq.(15) and Eq.(7), it gives the energy(mass) of the EPMD black holes in dS space-timeas M = b ( n − γ ( n − π ( α + 1) r α + n − α +1 c A ( x ) + q p r − p ( n + α − p − α +1) c B ( x ) , (21)where A ( x ) = A x − α − nα +1 − x α + n − α +1 (1 − x − α − nα +1 ) , B ( x ) = B x − α − p + n (2 p − α +1) − x − α − nα +1 (1 − x − α − nα +1 ) . (22)Taking the EPMD dS space-time as a thermodynamic system, in Refs. [1, 3, 12, 16, 50, 67], the thermodynamicvolume of the EPMD dS space-time is given by V = V c − V + . (23)Considering the correlation of BEH and CEH, we assume that the entropy of the EPMD dS space-time is expressedas S = S c + S + + S AB = b ( n − γ r ( n − − γ ) c x ( n − − γ ) + f AB ( x )] = b ( n − γ r ( n − − γ ) c F n ( x ) , (24)where f AB ( x ) is an arbitrary function of x .Using the effective thermodynamic quantities, the state parameters of the thermodynamic system satisfy the formulaof the first law of thermodynamics, i.e., dM = T eff dS − P eff dV + Φ eff dQ, (25)where the effective temperature T eff , the effective pressure P eff and the effective potential Φ eff of the EPMD dSblack hole system are respectively defined as T eff = (cid:18) ∂M∂S (cid:19) Q,V = k ( n − n − b − γ π ( α − α + n − x (1 − x − α − nα +1 ) r α − α +1 c T ( x ) T ( x ) , (26)with T ( x ) = − (cid:20) ( α + n − x α + n − α +1 − x n − α α +1 ) + ( α − n ) x − α − nα +1 (1 − x α + n − α +1 ) (cid:21) + q p r − pα − p +2 np +2(2 p − α +1) c p p (2 p − ( α + n − α − b − n − pγ (2 p −
1) +2 γ Π( α + n − p )( n − k − ( α − p + n )( x − α − p + n (2 p − α +1) − x n − α α +1 )(2 p −
1) + ( α − n ) x − α − nα +1 (1 − x − α − p + n (2 p − α +1) ) , (27) T ( x ) = F n ′ ( x )(1 − x n − α α +1 ) + x n − − α α +1 ( n − − γ ) F n ( x ) . (28)When T = T + = T c , which means the dS black hole thermodynamic system including BEH and CEH are thermo-dynamically equilibrium, the effective temperature of the space-time should be equal to the radiation temperatures ofBEH and CEH so as to it is with thermodynamic significance. So, when the charge of the space-time q p is expressedby Eq.(18), T eff = ˜ T eff = T + = T c . Substituting Eq.(18) into Eq.(26), it gives˜ T eff = − (1 + α ) ( n − kb − γ ( n − − α ) α − n )( α + n −
2) (1 − x α + n − α +1 )(1 − x − α − nα +1 ) − A ( x ) B ( x ) − ( α − n )(2 p − α + n − p ) (1 − x − α − p + n (2 p − α +1) )(1 − x − α − nα +1 ) = ˜ T ( x ) T ( x ) . (29)where ˜ T ( x ) = k ( n − α + 1) b − γ ( α − x (1 − x − α − nα +1 ) ˜ T ( x ) , (30)˜ T ( x ) = − ( α + n − x α + n − α +1 − x n − α α +1 ) − ( α − n ) x − α − nα +1 (1 − x α + n − α +1 ) + A ( x ) B ( x ) ( α + n − p − α + n − p ) − ( α − p + n )[ x − α − p + n (2 p − α +1) − x n − α α +1 ](2 p −
1) + ( α − n ) x − α − nα +1 [1 − x − α − p + n (2 p − α +1) ] . (31)Comparing Eq.(29) with Eq.(20), one can get T ( x ) = ( n − x n − α +1 + x n − α α +1 )(1 + α ) x (1 − x − α − nα +1 ) = ( n − − γ ) x n − − nγ + γ (1 + x n +1 − nγ − γ )(1 − x n − γ ( n +1) ) . (32)From Eq.(32) and Eq.(28), a differential equation of F n ( x ) can be obtained. Taking the initial condition as F n (0) = 1,ie., f AB (0) = 0, which indicates that the interaction between the two horizons is zero when x = 0, the solutions ofthe differential equation are F n ( x ) = 3( n − γn ) − γ − n − γn ) − − x ( n − γn − γ ) ] ( n − − γ ) / ( n − γn − γ ) − ( n − γn − γ )[1 + x n − γn ) − ] + [2( n − γn ) − − x n − nγ − γ − x n − nγ − )[2( n − γn ) − − x ( n − γn − γ ) ] + 1 + x ( n − − γ ) = f AB ( x ) + 1 + x ( n − − γ ) . (33)Substituting Eq.(33) into Eq.(26), the effective temperature of the EPMD dS space-time can be expressed as T eff = k ( n − b − γ T ( x )4 πr − γc ( α − α + n − − γ ) x n − − nγ + γ (1 + x n +1 − nγ − γ ) . (34) V. ENTROPIC FORCE BETWEEN THE TWO HORIZONS IN THE EPMD DS SPACE-TIME
The definition of the entropic force in the thermodynamic system is [24–37] F = − T ∂S∂r , (35)where T is the system temperature and r is the system radius. From Eq.(24), the entropy created by the interactionbetween BEH and CEH is S AB = b ( n − γ r ( n − − γ ) c f AB ( x ) . (36)According to the expression Eq.(35), the corresponding entropic force between the two horizons can be given as F = − T eff (cid:18) ∂S AB ∂r (cid:19) T eff , (37)where T eff is the effective temperature of the system and r = r c − r + = r c (1 − x ). Then F ( x ) = − k ( n − b ( n − γ r ( n − − γ ) c T ( x )16 π ( α − α + n − − γ )( n − − γ ) f AB ( x ) ddx h T ( x ) x n − − nγ + γ (1+ x n +1 − nγ − γ ) i + T ( x ) f AB ′ ( x ) x n − − nγ + γ (1+ x n +1 − nγ − γ ) x n − − nγ + γ (1 + x n +1 − nγ − γ )(1 − x ) ddx h T ( x ) x n − − nγ + γ (1+ x n +1 − nγ − γ ) i − T ( x ) . (38)In order to describe the behaviors of the entropic force created by the interaction between BEH and CEH andthe effect of the parameters of the MPMD dS space-time on the entropic force, the solutions for the entropicforce F ( x ) with different parameters, n , α , p and κ are depicted in the following figures, where we have taken q p r − p ( n + α − p − α +1) c b − n − pγ (2 p −
1) +2 γ = κ , k ( n − b ( n − γ r ( n − − γ ) c π ( α − α + n − − γ ) = 1 and k = 1. n = n = n = x - - F ( x ) FIG. 1: F ( x ) − x curve with different values of the pa-rameter n for α = 0 . p = 1 . κ = 0 . α = α = α = x - - - F ( x ) FIG. 2: F ( x ) − x curve with different values of the pa-rameter α for n = 3, p = 1 . κ = 0 . If the effect of entropic force is same to that of normal forces, from the F ( x ) − x curves, it is clear that the entropicforce tends to infinity with x → p = p = p = x - - - F ( x ) FIG. 3: F ( x ) − x curve with different values of the pa-rameter p for n = 5, α = 0 . κ = 0 . κ = κ = κ = x - - F ( x ) FIG. 4: F ( x ) − x curve with different values of the pa-rameter κ for n = 3, α = 0 . p = 1 . two horizons will separate from each other due to the entropic force with a corresponding large acceleration providedthe other forces are absent. This agrees with the present viewpoint on early cosmic inflation. When the value of x reduces from 1, the entropic force between the two horizons is decreasing until it reaches a minimum, and at x = x in the interval, the entropic force is zero, which can be interpreted as that the interaction between the two horizonsis absent at x = x , however where the two horizons may keep separation state. When the value of x reducesgradually, the value of the entropic force keeps in a negative territory temporarily, which means that the separationof the two horizons is decelerated and can be interpreted as the cosmic decelerated expansion. If the expansion speedis decelerated to zero before x reach to minimum at x = x , the two horizons will be in a relative oscillatory motionwith the equilibrium position x = x under the circumstance of no other forces exist. In the four F ( x ) − x figures,when x reduces from x = x , the value of the entropic force tends to zero in the negative territory before it goesto positive value (meaning repulsive force) at x = x except the blue dotdashed curve in FIG.4. In the region ofsmaller x , the behaviors of the entropic force are complicated, most of curves in the four F ( x ) − x figures go throughtheir singularities and go from positive to negative with decreasing x . but the singularity disappears on the blacksolid curves in FIG.2, which corresponds to the situations of smaller n with n = 3, smaller α with α = 0, smaller p with p = 1 .
3, and smaller κ with κ = 0 . κ is bigger the behavior of the entropic force isdifferent, which can be seen from the blue dotdashed curve in FIG.4. These situations of F ( x ) − x curves indicate thatthe behavior of entropic force is affected by the parameters n , α , p and κ , that is, it is influenced by the dimensionof the space-time, the nonlinearity of the electromagnetic field, the strength of coupling of the dilaton scalar andelectromagnetic field, the position of cosmological horizon, and the electric charge of the black hole. In all these casesof the four figures, the behaviors of the entropic force near x = x , or in the region of 1 > x > x are similar to that ofLennard-Jones force between two particles [71–73]. They are similar but obtained by completely different ways. Thisindicates that the entropic force between the two horizons has a certain internal relationship with Lennard-Jones forcebetween two particles. What is the fate of the accelerated expanding universe, whether the entropic force betweenBEH and CEH is one of the participant forces which drive the evolution of the universe, and whether the entropicforce has the same effect with Lennard-Jones force, which need more investigations and more evidences. VI. CONCLUSION AND DISCUSSION
The entropy of the charged dilaton black holes with Einstein power-Maxwell field in dS space-time is derived anddiscussed in the paper considering the correlation between BEH and CEH, especially the correction term caused bythe interaction between BEH and CEH. The entropic force F ( x ) between BEH and CEH is deduced according tothe definition of the entropic force in thermodynamic system. We discuss the entropic force F ( x ) changes with x ,the position ratio of BEH and CEH, in the EPMD dS space-time when the the parameters n , α , p , and κ take somecertain values. It is found that the behaviors of the entropic force F ( x ) at a larger interval of x are similar to that ofLennard-Jones force between two particles, and in a smaller interval of x the behaviors of the entropic force F ( x ) arecomplicated, which are related to the parameters of the space-time.Comparing the F ( x ) − x curves in a large interval of x with the curve of Lennard-Jones force versus the distanceof two particles given in reference[71–73], we find that the two curves are very similar although they are obtained indifferent ways. The entropic force between the two horizons is completely derived from general relativity. But theLennard-Jones force between two particles is concluded from simulation based on experiments. This indicates thatthere may be a relationship between the entropic force and the Lennard-Jones force.According to modern cosmology, the fate of our universe is dominated by matter and energy. If there are enoughmatter and energy, the gravitational effect will stop the cosmic expansion at a certain time, and then our universewill turn to contraction. Otherwise, if the density of cosmic matter and energy is too low, the universe will expandforever. Different kinds of cosmic matter and energy play different roles in the cosmic expansion. In this work, wefind that the entropic force probably plays a certain role in the cosmic expansion and contraction, other than cosmicmatter and energy. In the EPMD dS space-time, the influence of different parameters on the entropic force betweenBEH and CEH is shown in the F ( x ) − x curves. This indicates that different parameters in the EPMD dS space-timeplay different roles in the cosmic expansion.Since the space-time and the thermodynamic effect are relative to general relativity and quantum mechanics re-spectively, and all physical quantities satisfy the first law of thermodynamics. If the effect of entropyic force is provedsimilar to that of normal forces, it indicates that there is a relationship among general relativity, quantum mechanicsand thermodynamics. It will provide a new way to study the interaction between particles in black holes, the mi-crostate of particles in black holes, the Lennard-Jones potential between particles and the microstate of particles inordinary thermodynamic systems. Acknowledgments
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