Holographic heat engine with momentum relaxation
aa r X i v : . [ h e p - t h ] O c t Holographic heat engine with momentum relaxation
Li-Qing Fang ∗ School of Physics and Electronic Information,Shangrao Normal University, Shangrao 334000, China.
Xiao-Mei Kuang † Center for Gravitation and Cosmology, College of Physical Science and Technology,Yangzhou University, Yangzhou 225009, China
Abstract
We investigate the heat engine defined via black hole with momentum relaxation, which is introducedby massless axion fields. We first study the extended thermodynamical properties of the black holeand then apply it to define a heat engine. Then, we analyze how the momentum relaxation affectsthe efficiency of the heat engine in the limit of high temperature. We find that depending onthe schemes of specified parameters in the engine circle, the influence of momentum relaxation onthe efficiency of the heat engine behaves novelly, and the qualitative behaviors do depend on thedimension of the gravity theory. ∗ Electronic address: [email protected] † Electronic address: [email protected]
I. INTRODUCTION
Thermodynamics of the anti de Sitter (AdS) black holes have been attracting more and more attention inthe development of holographic gauge/gravity duality, because they are important for us to understand thenature of quantum gravity. Recently, by treating the cosmological constant as the pressure of the black holethermal system[1–4], the study of black hole thermodynamic has been extended into a more general case,in which the thermodynamical volume is defined as the conjugate variable of the pressure, and the mass ofthe black hole is considered as the enthalpy of the system. This proposal has inspired plenty of interestingresearches and remarkable phase structures of AdS black holes, please see Ref. [5] and therein for nice reviews.In the framework of the extended thermodynamic of the black hole, the idea of defining a traditional heatengine via an AdS black hole was proposed in [6]. In details, the heat engine is realized by a circle in thepressure-volume phase space of the black hole. The input of heat ( Q H ), the exhaust of heat ( Q C ) and themechanical work ( W ) can be evaluated from the black hole system. Since the engine circle represents a processdefined on the space of the dual field theory living in one dimension lower, so that this kind of engine definedvia an AdS black hole is named as holographic heat engine[6]. More remarkable properties of holographicheat engine were widely studied in black holes with Gauss-Bonnet correction[7], in Born-Infeld corrected blackhole[8], in rotational black hole[9, 10], in three dimensional black hole[11], and so on[12, 13].However, until now the holographic heat engines were mainly defined via the black holes dual to theboundary theory with translational symmetry, i.e. without momentum relaxation. It would be more realisticand interesting to explore the heat engine defined via black holes with momentum relaxation, because ourworld is far from being ideal. Thus, the aim of this work is to study how the momentum relaxation of theblack hole affects the efficiency of the related holographic heat engine.We will focus on the heat engine in the Einstein-Maxwell-Axions theory with negative cosmological constant,where the momentum relaxation is introduced by the linear massless axion fields[14]. We will mainly study theinfluence of momentum relaxation on the efficiency of the heat engine. Borrowing holography, it was foundthat the momentum relaxation brings many novel properties in the dual condensed matter sectors[15–18],because it modifies the bulk geometry and breaks the translational symmetry of the boundary theory in asimple way. So we should expect that it will enrich the properties of the related heat engine due to the duality.Meanwhile, the study of the heat engine will conversely help to further understand the thermal systems withmomentum relaxation and their holographic duality aspects.The remaining of this paper is organized as follows. We briefly review the AdS black hole solution inEinstein-Maxwell-Axions theory, and study the extended thermodynamics in section II. Then in section III,by giving two schemes of specified parameters, we will study how the efficiency of the heat engine is affectedby the momentum relaxation. Section IV contributes to our conclusions and discussions. II. REVIEW OF BLACK HOLE IN EINSTEIN-MAXWELL-AXIONS THEORY AND THEEXTENDED THERMODYNAMICS
The Einstein-Maxwell-Axions gravity theory was proposed in [14] by introducing D − S = 116 π Z d D x √− g R − − F µν F µν − D − X I =1 ( ∂ψ I ) ! , (1)where D is the dimension of the spacetime, and the cosmological constant isΛ = − ( D − D − L . (2)By setting the scalar fields to linearly depend on the D − x a , i.e., ψ I = βδ Ia x a , one finds that the action admits the charged black hole solution ds = − f ( r ) dt + 1 f ( r ) dr + r dx a dx a , A = A t ( r ) dt, with f ( r ) = r L − β D − − mr D − + q r D − , A t = r D − D − − r D − h r D − ! qr D − h (3)where the index a goes a = 1 , · · · D −
2, and the horizon r h satisfies f ( r h ) = 0. Note that the horizon hasflat topology and the solution is valid only if D >
4. It is worthwhile to point out that the scalar fields in thebulk source a spatially dependent field theory with momentum relaxation, which is dual to a homogeneousand isotropic black hole (3). The linear coefficient β of the scalar fields somehow can be considered to describethe strength of the momentum relaxation in the boundary theory[14].The mass and charge of the black hole are connected with the parameters m and q as M = ( D − V D − π m, and Q = p D − D − V D − π q (4)where V D − is the volume of the D − T = f ′ ( r h )4 π = 14 π ( D − r h L − β r h − ( D − q r D − h ! , (5)and the entropy is S = V D − r D − h . (6)We will study the thermodynamic in the extended phase space. As proposed in [1–4], we assume the relation P = − Λ8 π = ( D − D − πL . (7)Then the mass of the black hole can be rewritten as M = ( D − V D − π πP r D − h ( D − D −
2) + q r D − h − β r D − h D − ! (8) In general, the linear combination form of the scalar fields are ψ I = β Ia x a . Then defining a constant β ≡ D − ( P D − a =1 P D − I =1 β Ia β Ia ) with the coefficients satisfying the condition P D − I =1 β Ia β Ib = β δ ab , we will obtain the sameblack hole solution. Since there is rotational symmetry on the x a space, we can choose β Ia = βδ Ia without loss of generality. which is defined as the enthapy H of the system. Subsequently, the thermodynamical volume is V = ∂M∂P (cid:12)(cid:12) S,Q = V D − D − r D − h , (9)and the electric potential is Φ = ∂M∂Q (cid:12)(cid:12)
S,P = 16 πQ ( D − V D − r D − h = s D − D − qr D − h (10)where in the second equality, we used the relation in (4).
200 400 600 800 1000 r h P FIG. 1: P − r h diagram for four dimensional black hole with momentum relaxation. We fix T = 10 and increase β = 0 , ,
200 in the lines from the bottom to the top. In the plot, we set q = 1 and D = 4. It is straightforward to verify that the first law of thermodynamics is dM = T dS + Φ dQ + V dP + ϕdβ (11)where ϕ = ∂M∂β (cid:12)(cid:12) S,P,Q = − ( D − β V D − r D − h π ( D − . And we obtain the Smarr relation for the black hole is( D − M = ( D − T S + ( D − Q − P V. (12)Note that in (11), we see that β is a variable in the law, however, in the above Smarr relation, β andits conjugation ϕ do not have explicit contributions. This observation is reasonable because, after carefulcomparison, we find that it is consistent with the contribution of c term in massive black hole to the extendedthermodynamics [21].We then substitute the pressure (7) into the temperature (5), after which we can solve out the pressure as P = ( D − π πTr h + β r h + ( D − q r D − h ! . (13)The above expression is also treated as the state equation. From (13), we can figure out the P − r h or P − V diagram with fixed temperature and momentum relaxation. The results are shown in figure 1. It is notedthat here we are not interested in the existence of P − V criticality in the extended phase space proposed inRefs.[22–24]. We can also calculate the electric potential Φ, which is measured at infinity with respect to the horizon, via the methodaddressed in [19, 20], i.e., Φ = A t | r →∞ − A t | r → r h = q D − D − qr D − h . Here we have used the expression of A t in (3) andobviously we obtain vanishing static electric potential at the horizon. III. HOLOGRAPHIC HEAT ENGINE VIA BLACK HOLE WITH MOMENTUM RELAXATION
In this section, we intend to define kind of heat engine via the black hole with momentum relaxationdescribed in last section.
A. The general engine efficiency
Before studying the efficiency of the engine, we have to calculate the specific heat
T ∂S/∂T , which can becomputed from the expressions for temperature and the entropy. We treat both T and S as functions of thehorizon r h . Then from (6), we obtain ∂S∂r h = ( D − V D − r D − h . (14)Differentiation of the state equation (5) gives us dT = 14 π " πPD − β r h + ( D − D − q r D − h ! dr h + 16 πr h D − dP (15)from which we can reduce ∂T∂r h = πPD − + β r h + ( D − D − q r D − h π − πr h D − ∂P∂T . (16)Subsequently, we have the general formula of the specific heat C = T ∂S∂T = T (cid:16) ∂S∂r h (cid:17)(cid:16) ∂T∂r h (cid:17) = (cid:18) − r h D − ∂P∂T (cid:19) πD − P r D − h − β r D − h − ( D − q πD − P r D − h + β r D − h + ( D − D − q ( D − V D − r D − h . (17)From equation (9), it is obvious that constant volume means also constant r h . And equation (13) gives us( ∂P/∂T ) V = ( D − / r h , so the specific heat at constant volume is zero, C V = T ∂S∂T (cid:12)(cid:12) V = 0 . (18)While the specific heat at constant pressure, C P , can be computed by setting ∂P/∂T = 0 in equation (17) C P = T ∂S∂T (cid:12)(cid:12) P = πD − P r D − h − β r D − h − ( D − q πD − P r D − h + β r D − h + ( D − D − q ( D − V D − r D − h . (19)In order to define a heat engine, we will consider a rectangle cycle in the P − V plane as it was done inthe previous literatures[8–13] . The rectangle consists of two isobars and two isochores as shown in figure 2.The vanishing of C V describes the “isochores” while the heat flow along the isobars can be evaluated if C P is determined. Note that 1 , , , , , , W = ( V − V )( P − P ) , (20) VP FIG. 2: Cartoon of the engine. and the input of the heat is Q H = Z T T C P ( P , T ) dT . (21)So the engine efficiency is η = W/ Q H . Note that in P − V plane, the isotherms at temperatures T h and T l with T h > T l give the Carnot efficiency η C = 1 − T l /T h . And for our engine, it is η C = 1 − T l T h = 1 − T T . (22)In what follows, we will study η for the heat engine and focus on the effect of momentum relaxation. Weshall first work in four dimensional theory, and then only show the results for five dimensional case in theappendix A. We will set q = 0 . V = 1 without loss of generality. B. Engine efficiency in large temperature limit
In order to evaluate the efficiency, we have to calculate the expressions (20) and (21). Usually, it is difficultto do the exact integration in (21). So we will consider the large temperature limit, i.e., T ≫ β, q , whichmeans that 1 /T can be treated as a small quantity . We first solve out r h in term of large T from (5), r h = T P + β πT + P (32 πP q − β )32 π T + · · · . (23)Then from (9) and (19), we can obtain V = 13 r h = T P + β T πP + q πT + · · · ,C P = T P + − πP q + β π T + P β (80 πP q − β )64 π T + · · · , (24)respectively. Note that the study with β = 0 goes back to that in Reissner-Nordstr¨ o m black hole but with spherical horizon in [6]. Here β can be arbitrary real constants in the black hole solution. However, in our study, β always appears in the form β , so we willfocus on real positive β to see its effect without loss of generality. Moreover, we will study the heat engine efficiency in largetemperature limit with T ≫ β, q . It is also interesting to study the effect of more general β . Consequently, the efficiency of the engine can be obtained η = W Q H = (cid:18) − P P (cid:19) × P ( V − V ) R T T C P ( P , T ) dT ≃ (cid:18) − P P (cid:19) × P ( T − T ) + β πP ( T − T ) P ( T − T ) + · · · ≃ (cid:18) − P P (cid:19) (cid:18) β π P T + T T + T + · · · (cid:19) (25)where the volumes V and V correspond to the volume for T and T at P = P = P , respectively.We shall disclose how the momentum relaxation β affects the efficiency of our heat engine by evaluating theratio η/η C and η/η , where η is the efficiency at β = 0. We will consider two schemes of specified parametersin the engine circle as pioneerly addressed in [7]. Note that in order to make sure that the dual heat engine isthermodynamically physical, we have to set the proper parameters to satisfy the efficiency lower than unit. Β Η (cid:144) Η C Β Η (cid:144) Η FIG. 3: Results in the fist scheme with D = 4. Left: the ratio η/η C . Right: the ratio η/η . We have chosen thespecific parameters of the cycle: P = 5 , P = 3 , T = 50 , T = 60. Firstly, we choose the engine cycle with specified ( T , T , P , P ), i.e. we set P = 5 , P = 3 , T = 50 and T = 60 for the engine in figure 2. Recalling the state equation (13), we can first calculate V through thegiven ( T , P ), then compute the temperature of heat source T l = T through the known ( P , V = V ) . Itis obvious that in this scheme T decreases as β increases, such that the Carnot efficiency η C = 1 − T /T isenhanced by β . On the other hand, we see from (25) that the efficiency η also increases when we increase themomentum relaxation. So, it is not direct to say how the ratio η/η C changes as β . We show the results infigure 3. The left plot shows that the efficiency is lower than the Carnot efficiency and the ratio η/η C increaseswhen momentum relaxation becomes stronger. While, the right plot shows that the efficiency is higher forbigger β , which is explicit from equation (25) with fixed ( T , T , P , P ).Let us turn to study the engine efficiency with specified ( T , T , V , V ). In this scheme, the Carnot efficiencywill not change with β . Similarly, we recall the state equation again. We can determine P = P and P via( T , V ) and ( T , V ), respectively. Then we calculate T via ( V = V , P ). Our results of the efficiency in thisscheme are shown in figure 4. The tendencies of η/η C and η/η are both suppressed by larger β , which are incontrast to those occur in the first scheme.The results of the efficiency for D = 5 are shown in appendix A. It is obvious that for the second scheme,the behaviors of η/η C and η/η depending on β are qualitatively the same as those in D = 4 case. However,for the first scheme, the ratio η/η C deceases as β increases, which is different from the behavior in D = 4dimensional theory. Β Η (cid:144) Η C Β Η (cid:144) Η FIG. 4: Results in the second scheme with D = 4. Left: the ratio η/η C . Right: the ratio η/η . We have chosen thespecific parameters of the cycle: T = 60 , T = 30 , V = 3000 , V = 1500. We work in D = 4. IV. CONCLUSION AND DISCUSSION
In this paper, we extended the study of the heat engine by defining it via a simple black hole with momentumrelaxation β . We especially studied the effect of momentum relaxation on the efficiency of heat engine inlarge temperature limit. Momentum relaxation introduced by linear massless axion fields has been widelystudied in holographic condensed matter topics, because it breaks the translational symmetry of the dualfield theory and gives finite conductivity in a simple way. β in this model modifies the state equation in theextended thermodynamics of the black hole, so that it has print on the heat capacity. Thus, the study ofheat engine affected by momentum relaxation may bring insights to both the dual boundary theory and thethermodynamics of the black hole.In two different schemes with specified parameters, we evaluated the efficiency η comparing to both theCarnot efficiency η C and the efficiency with β = 0, denoted η . In the first scheme with given ( T , T , P , P ),as β becomes stronger, both η/η C and η/η increases. In the second scheme with given ( T , T , V , V ), both η/η C and η/η decrease as β increases, which are different from those happen in the first scheme. The effectsof momentum relaxation on the efficiency for the second scheme in our model are opposite to the effectsof Born-Infeld correction in the Maxwell field in four dimensional case addressed in [8]. While for the firstscheme, the behavior η/η C obeys the same rule but η/η behaves in an opposite way. Moreover, comparingto the effect of Gauss-Bonnet correction[7], here the effect of β also qualitatively depends on the dimension ofthe gravity theory. In five dimensional thoery, η/η increases in both theories while η/η C behaves oppositelyin the first scheme, but in the second scheme, both η/η C and η/η decrease as Gauss-Bonnet coupling ormomentum relaxation increases. In our previous paper [25], we found the black hole solution in arbitrarydimensional charged Gauss-Bonnet-Maxwell-Axions theory, so it would be very interesting to study the mixedeffects of Gauss-Bonnet coupling and momentum relaxation, from which we may see the enhance/competitionof increasing the efficiency of the holographic heat engine. We shall report the results in the near future.The gravity theory we chose to study the heat engine is a simple theory dual to field theory with momentumrelaxation. A more general homogeneous theory without translational symmetry is the massive gravity pro-posed in [26, 27]. The extended thermodynamics of the massive black holes has been investigated in [21, 28].Very recently, the efficiency of the heat engine via black holes in massive gravity was addressed in [29, 30]where the authors studied the effects of more general bulk parameters on the efficiency of the heat engine.Note that both our studies of the effects of momentum relaxation on the heat engine focused on four or fivedimensional background. It would be interesting to study the heat engine via three dimensional black holeswith momentum relaxation, for example the BTZ massive black hole constructed in [31]. Acknowledgements
This work is supported by the Natural Science Foundation of China under Grant Nos. 11505116 and11705161. L. Q. Fang is also supported by Natural Science Foundation of Jiangxi Province under Grant No.20171BAB211013. X. M. Kuang is also supported by Natural Science Foundation of Jiangsu Province underGrant No. BK20170481.
Appendix A: Results in five dimensional theory
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