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II. MICROSCOPIC INTERPRETATION OF 0.5 H1, 0L 0.0 Ruppeiner observed for various fluid systems that a 0 Φ negative/positive curvature scalar of the geometry was  related to a respective attractive/repulsive interaction Φ -0.5 of the fluid molecules [18]. Employing this property, 1 Ruppeiner geometry has been used to examine the mi- J2 6 , -1N -1.0 crostructure of some complex fluid systems and black holes. Why this empirical observation works is still a 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 mystery. Since Ruppeiner geometry is closely related to rr0 phase transitions, we begin by providing a microscopic interpretation of Ruppeiner geometry for the VdW fluid FIG. 1: The Lennard-Jones potential. The red dashed curve model. and blue solid curve are for the short-range repulsive interac- Comparing to an ideal fluid, one needs to take into tion and longer-range attractive interaction, respectively. account the size of and interaction between the molecules for the VdW fluid, which are described by the parameters b and a, respectively. The equation of state reads freedom are left, and thus one may conjecture there is no interaction between these molecules. Furthermore, we 2 can expect that when this condition is violated repulsive NT N a T a 27 P = 2 = 2 , (2) interactions will be present. Second, when T < 32 Tc = V Nb − V v b − v a − − 4b , the pressure is negative for a given range of v, which where P , T , V , and v are the pressure, temperature, may lead to the repulsive interaction. total volume, and specific volume of the system, and N Making use of fluctuation theory, Ruppeiner geometry corresponds to the number of the molecules contained in has been used to great success in revealing the intrin- the fluid. The equation of state shows a first-order phase sic microstructure of black holes. In particular, the nor- transition, and admits a second order phase transition malized Ruppeiner curvature scalar was introduced and a 8a c c c 2 point at the critical point (P , T , v )=( 27b , 27b , 3b). shown for black holes to have universal behaviour near Such phenomena can be understood from a microscopic a critical point [20, 21]. Here we consider the relation viewpoint. As is well known, the interaction between two between the interactions and the normalized curvature fluid molecules is given by the Lennard-Jones potential scalar, which for a VdW fluid is given by [23] 2a(v b) ab + P v3 12 6 RN = − 2 . (5) r0 r0 − (2ab av + P v3) φ =4φ0 (3)  r − r −      Obviously, the denominator of RN is non-negative, van- illustrated in Fig. 1. The red dashed curve and blue ishing when the system approaches the critical point, −2 solid curve are for the short-range repulsive interaction which leads to RN (Tc T ) . Here we are concerned ∼ − and longer-range attractive interaction, respectively. At with the numerator of RN. Noting that the parameters a r = r0, we have φ=0, with the minimum of the potential and b are positive, we see that RN < 0 when v b. This 1 ≥ φ0 occurring at rmin =2 6 r0. A VdW fluid is described implies an attractive interaction, which is consistent with by− a “hard-core” model, meaning that the interatomic the microscopic potential model of the VdW fluid. distance between molecules cannot be smaller than a We can gain further insight by relaxing the require- molecular diameter d = rmin. Hence there is a cutoff ments on these parameters. First, we see that RN van- at the location of the well, and the short-range repulsive ishes when a = 0. Since a is a measure of molecular interaction denoted by the red dashed curve is excluded. interaction, vanishing a indicates there is no interaction, Furthermore, taking the mean field approximation, the consistent with empirical observations of Ruppeiner ge- characteristic parameters a and b in the equation of state ometry. As expected, from Fig. 1 and Eq. (4), pos- (2) have a microscopic interpretation [24] itive/negative a corresponds to a well/barrier, indicat- ing attractive/repulsive interactions respectively. Since 3 20πr0φ0 3 RN has opposite sign to a, a negative/positive curva- a = , b = √2r0. (4) 9√2 ture scalar indicates respective attractive/repulsive in- teractions. The absolute value of RN provides a measure where a > 0, indicating the attractive interaction domi- of the strength of these interactions. nates the fluid microstructure. Turning to the second term (v b), if b = v, all the We now clearly address two issues for a VdW fluid space of the fluid system is occupied− by the molecules, so that are often neglected. The first one is the volume no interactions are allowed. The corresponding curvature v b. When v = b, it means the molecules completely scalar vanishes, consistent with empirical observations of occupy≥ the space of the system. No spatial degrees of Ruppeiner geometry. If we do not impose a cutoff at 3 the location of the well in the Lennard-Jones potential, 0.05 shifting it to smaller r, then a short-range repulsive in- 0.00 teraction is present. There is a range of negative (v b), -0.05 − and RN becomes positive as expected. 3 -0.10

3 0 Finally, considering the term (ab + P v ), it is positive r since a fluid system has positive pressure and volume. Φ* -0.15 a -0.20 But we have noted that when T < 4b , the pressure be- comes negative, yielding positive RN. This case also pro- -0.25 vides an opportunity for repulsive interactions, and is a -0.30 combined effect of both a and b. 0.0 0.5 1.0 1.5 2.0 The VdW parameter space has been shown to have a rr0 region where RN is positive [21], but it completely falls within the coexistence region. It was therefore excluded FIG. 2: The molecular potential (12) for the Schwarzschild since it was not clear that the equation of state was ap- AdS black hole. plicable in the coexistence region. Only the attractive in- teraction then dominates in the whole parameter space, from which we can write which is also consistent with the microscopic model of a 2 VdW fluid. However if the equation of state is applicable NT N T 1 P = = (10) in this region, the positive region of RN is actually caused V − 2πV 2 v − 2πv2 by negative pressure. for the equation of state for the Schwarzschild AdS black In summary, combining with the Lennard-Jones poten- hole. Comparing with the equation of state of the VdW tial, we have given a clear microscopic interpretation of fluid (2), we can immediately obtain Ruppeiner geometry for a VdW fluid. We shall now ap- ply this interpretation to black hole systems, constructing 1 a = , b =0. (11) the corresponding molecular potentials. 2π After a simple calculation, we give a possible effective III. SCHWARZSCHILD ADS BLACK HOLE potential between two Schwarzschild AdS black hole molecules In an AdS space, Schwarzschild black hole is the most 495 1 1 φ = ,(12) simplest black hole solution, which is described by the 2 3 6 12 6 6 32√2π r0 √ − √ following action " r/r0 + 2 r/r0 + 2 # 0 1 4 6 which is plotted in Fig. 2. Here the parameter r has S = d x√ g R + , (6) 16πG − l2 the dimension of length. Clearly, this gives an attractive Z   interaction for r/r0 0. At r/r0 = 0, one has minimum 3 ≥ where l is the AdS radius and it is interpreted as the φmin 0.28/r0. This pattern is consistent with that 2 pressure of the black hole P =3/(8πl ) [16]. Solving the of the≈ normalized − scalar curvature given in Ref. [25], corresponding field equations, the line element reads where only the attractive interaction dominates for the

2 2 1 2 2 2 Schwarzschild AdS black hole. ds = f(r)dt + dr + r dΩ2, (7) − f(r) where the metric function f(r)=1 2M/r + r2/l2 and IV. CHARGED ADS BLACK HOLE M denotes the black hole mass. Requiring− the absence of conical singularities at the horizon in the Euclidean sec- Four-dimensional charged AdS black holes were the tion of the black hole solution, the Hawking temperature first ones found to possess a small-large black hole phase can be easily obtained transition [14], reminiscent of the liquid-gas phase transi- tion of a VdW fluid. The line element is still of the form ∂rf(rh) 1 2 2 T = = +2P rh (8) (7), but with metric function f = 1 2M/r + Q /r + 4π 4πrh r2/l2, with Q being the black hole− charge. Compared with rh the radius of the black hole horizon, which is with the Schwarzschild AdS black hole, the existence of actually the equation of state of the black hole sys- the phase transition is mainly caused by the present of tem. It is easy to check that the first law holds, i.e., the black hole charge. The equation of state is dE = T dS P dV , with the respective entropy and ther- − 2 4 3 NT N 2 2N 4Q2 T 1 2Q2 modynamic volume S = πrh and V = 3 πrh. Noting that 2 P = 2 + 4 = 2 + 4 . (13) the specific volume v = 2rhlP [14], we have the number V − 2πV πV v − 2πv πv of the potential black hole molecules A phase transition exists caused by the charge. Regard- V 2 2 2 ing the black hole system as being composed of under- N = = πrh = S, (9) v 3 3 lying molecules [19, 20], the charge must be an intrinsic 4

property of these molecules. It is then natural to define 0.0000 an effective specific charge Q -0.0001

q = N , (14) Q ∗ V

Φ* -0.0002 from which the equation of state (13) can be cast into a new form -0.0003

T 1 2 -0.0004 P = 2 (1 4q ). (15) 0 2 4 6 8 10 v − 2πv − 1 rQ 3 Comparing with (2) of the VdW fluid, we obtain FIG. 3: The molecular potential (17) for the charged AdS 1 2 a = (1 4q ), b =0. (16) black hole after fitting the local well with the zero point of 2π − the normalized curvature scalar [20, 21]. When the black hole charge vanishes, q=0, reducing to (11) for the Schwarzschild AdS black hole case. The charged AdS black hole microstructure can be V. SUMMARY dominated by repulsive interactions for small black holes [20]. Following the microscopic interpretation of Rup- peiner geometry, this can be understood in terms of By combining empirical observations Ruppeiner geom- a Lennard-Jones like potential that is not cut off at etry with the microscopic model of a VdW fluid with the well. A short-range repulsive interaction is present the Lennard-Jones potential, we have obtained the first that will lead to a dominant repulsive interaction after microscopic interpretation for the constituent degrees of a mean-field approximation. When the charge is in- freedom of a (charged) AdS black hole. We have given for cluded, one can construct the corresponding potential the first time explicit molecular potentials for these con- from (12). However as we know, the molecular poten- stituents, successfully implementing the potential black tial is not unique for the same equation of state. Here we hole molecule model proposed previously [19, 20]. Our would like to model the potential with a new interesting results address the outstanding problem of why attrac- form tive/repulsive interactions between the microscopic con- stituents of a empirically corre- 12 6 r0 r0 sponding negative/positive Ruppeiner curvature scalars. φ =4φ0 , (17) r + r0 − r + r0 "    # Our results suggest that the repulsive interaction comes from two aspects. One is the finite size of the where the parameter r0 is molecular constituents; the other is the cutoff of the 2 well, where the short-range repulsive interaction should 3 495 1 4q be present in the potential. In particular, we found that r0 = −2 , (18) 248 π φ0  a repulsive interaction is present for the charged AdS black hole whilst absent for the Schwarzschild AdS black which clearly depends on the black hole charge. For this 1 6 hole. Note that the molecules have a finite actual size af- case, the well is located at rmin = (2 / 1)r0. When − ter an approximate comparison with the VdW fluid [26], 0

Acknowledgements black hole molecules for a dyonic black hole. This work was supported by the National Natural Science Founda- The authors would like to thank Prof. Hong Zhao for tion of China (Grants No. 12075103, No. 11675064, No. useful discussions. As this paper was being completed, 11875151, and No. 12047501), the 111 Project (Grant we became aware of Ref. [22], in which the configuration No. B20063), and the Natural Sciences and Engineering integral was used to propose an interaction between the Research Council of Canada.

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