Wei Sun(孫威) and Xian-Hui Ge(葛先輝)

Department of Physics,Shanghai University,Shanghai 200444,China

Keywords: holographic heat engines,benchmark cycle,hyperbolic charged black holes

1. Introduction

Recently, some authors consider the cosmological constant as the pressure in the thermodynamics of black holes,[1–4]which has led to many interesting related studies. For example, some interesting black hole thermodynamic phenomena and rich phase structures similar to the van der Waals fluids are discovered.[5]Other authors have built a bridge between extended black hole thermodynamics and quantum complexity.[6]In 2014,Johnson proposed a concept of a holographic heat engine by considering AdS black holes as working matter.[7]We show a schematic diagram of the usual heat engine in Fig.1. In general,the heat engine efficiency can be expressed as

whereQHandQCare the heat from the high-temperature source to the heat engine and the heat from the heat engine to the low-temperature source,respectively. The efficiency of the heat engine depends on the path selection on theP–Vdiagram. For the Carnot cycle,the path consists of two isotherms and two adiabatic lines on theP–Vdiagram,and the change of entropy is zero. The efficiency of the Carnot cycle with temperatureTHandTCisηC=1?TC/TH,which is the maximum efficiency that can be achieved. However, theP–Vdiagram of the AdS black hole is complicated, and difficult to establish the Carnot cycle. Johnson proposed a rectangular cycle,which consists of two equal-pressure processes and two equalvolume processes(Fig.2). The rectangular cycle is equivalent to the Carnot cycle of a static black hole,because in this case,both entropy and volume depend on the single variable of the horizon radiusr+. In other words,we can replace the volume on the horizontal axis in Fig.2 with entropy. In this case,the specific heat capacity of a constant volume is zeroCV=0,the equal-volume process is the same as the adiabatic process.The specific heat capacity of a constant pressureCPcan be calculated by the radius of the horizon.QHandQCcan be expressed as[8]

The index numbers in the subscript represent the four stages in the cycle shown in Fig.2. The heat engine efficiency is

In the AdS black hole thermodynamics,mass is defined as enthalpy dH=TdS+VdP. In the rectangular cycle, the pressure remains constant during the heat exchange process,i.e.,dP=0. CombinedQHwithQC, the heat exchange can be regarded as the change of the black hole massM,

whereM1,M2,M3, andM4respectively correspond to the black hole mass in the four stages of the cycle. Johnson proposed an exact efficiency formula for the holographic black hole heat heat engines[8]

Fig.1. The usual heat engine.

Fig.2. The usual heat engine.

The efficiency of black hole heat engine is an important part of the study of black hole thermodynamics. However,most authors only discuss the case with the horizon being a positive constant curvature hypersurface.[9–22]Note that there are huge differences in the thermal stability of black holes in AdS spaces with different curvatures. It is also necessary to study the heat engine efficiency of the AdS black holes with negative constant curvature horizon. In fact, some authors have studied the thermal stability of hyperbolic black holes.[23]Based on the study of thermodynamics of hyperbolic black holes, in this paper we further discuss the efficiency of the holographic heat engine of hyperbolic black holes. In the next section we briefly review its thermodynamics, then consider charged hyperbolic black holes as working matter to establish a rectangular heat engine cycle, and further discuss the effect of electric charge and the efficiency of the flat case and spherical case. In Section 3, we calculate the efficiency of benchmark cycle by using numerical simulation, and make a summary and discussion in the last section.

2. Hyperbolic charged black hole as a heat engine

The metric of the(n+2)-dimensional hyperbolic charged AdS black hole is

where

wherek=1,0,?1 respectively describe the spherical case,flat case,and hyperbolic case. If no special instructions are given,we mainly focus on the casek=?1 in the following.mand ?qare two integration constants, dΣ2nstands for the line element for ann-dimensional hypersurface with negative constant curvature?n(n ?1). The authors of Ref. [24] found that the hyperbolic black hole is supersymmetric in the case ofm=0 and ?q=l2/4, in which the metric can be expressed asf=(l/2r ?r/l)2,which is an extremal black hole solution with vanishing Hawking temperature and the mass parameter is zero. Whenm=0,there are still two black hole horizons

We can see that this efficiency is the same as the result obtained by Johnson in Ref. [8]. Next, we will compare the efficiency of the hyperbolic black hole with the well-known Carnot efficiency. We rewrite the temperature(Eq.12)and the entropy(Eq.13)as

The efficiency can be rewritten as

Fig.3. The isotherms and rectangular heat engine cycle.

Considering the isotherm shown in Fig. 3, we can find thatT2is the highest temperature(T2=TH)andT4is the lowest temperature (T4=TC). Note that whenS4→S2, the efficiency(23)can be written as

which is the efficiency of the Carnot cycle. In other words,when the entropy (volume) change is small enough (?S=S2?S1=S3?S4), the rectangular cycle becomes the Carnot cycle(As shown in Fig.4).

Fig. 4. The vertical axis in the figure represents the ratio of the hyperbolic black hole rectangular cycle efficiency η to the Carnot efficiency ηC, and the horizontal axis is the entropy S4. The blue line represents the heat engine efficiency of the four-dimensional hyperbolic black hole when ?q=1.We set Ω =1,T2=5,T4=1,and S2=8.

Furthermore, we can obtain the efficiencies of spherical(k=1) and flat (k=0) case in the same way. The results(as shown in Fig.5(a))show that the efficiency of hyperbolic black holes are lower than the cases ofk=1 andk=0. Note that we restricted ourselves to the case ?q>0,since when ?q=0,the black hole withk=0,?1 is not physical. In addition,figure 5(b)shows the hyperbolic black hole effect with different electric charges.The results show that less charge corresponds to higher efficiency in the case of ?q>0, because the charge confines part of the energy.

Fig. 5. (a) Black hole heat engine efficiency with different k, where we choose ?q=0.1.(b)The efficiency with different electric charges ?q=1,5,10.

3. Benchmark cycle for hyperbolic charged black hole

In fact, since the rectangles can be added, we can use equation(6)as a basis to calculate any closed loop on theP–Vgraph[8](as shown in Fig. 6). Note that only the upper and lower edges of the rectangle will generate heat flow.

In this section, we introduce a benchmark cycle,[25,26]which could be parameterized as a circle. We cover a circle with a center of(P0,V0)and radiusRonN×Nregular lattice of squares,whereNis an even number.QHis determined by the sum of the mass differences at both ends of each“hot cell”isobar. The cell refers to the small square in Fig.6. Edge cells are called hot cells if they have their upper edges open, and cold cells if they have their lower edges open.QCis determined by“cold cell”.

Fig.6. Adding cycles that share an edge.

A more general expression of efficiency can be obtained through the first law of thermodynamics. Consider for convenience a circular benchmarking cycle defined parametrically by the following expressions:

The first law of thermodynamics in the canonical ensemble can be expressed as

The first law can help us to express the integral ofQHandQCin the process by the area of the circle and the difference in mass (see Fig. 7). The area of the circle is divided into two parts byQ1andQ2,and two lines parallel to theVaxis divide the circle into three parts,A1,A2,andB(B=πR2?A1?A2).Then we can get theQHandQC[27]

Efficiency can be expressed as

In particular, when the specific heat capacity at constant volume vanished(CV=0),the above equation can be rewritten as[19]

where ?Mis the enthalpy difference of two points at the left and right ends of the circle

Based on the thermodynamic quantities of the charged hyperbolic black hole, we have obtained the four-dimensional heat engine efficiency by numerical simulation. We choose the circle whose origin is at (P0,V0)=(20,110) and the radius isR=10(shown in Fig.7(b)).

Fig. 7. (a) Example of covering a circle on 100 regular lattices of squares(N=10). The origin is at(P0,V0)=(20,110)and the radius is R=10. The red line represents the top edge of the hot cell,and the blue line represents the bottom edge of the cold cell. As N increases,these lines converge to the boundary of the circle. (b)A general benchmarking cycle, partitioned into subregions.

Fig.8. The efficiency of benchmarking cycle as a function of regular lattice of squares N. For N =5×104, η converge to 0.56704. Here the origin is(P0,V0)=(20,110)and the radius is R=10,we take ?q=0.5 and G=1.

The efficiency of charged hyperbolic black holes (as shown in Fig. 8) converges to 0.56704. In fact, we can also rewrite temperature as a function of pressure and volumeT(P,V), the efficiency of the Carnot cycle isηC=0.66677.We can see that the efficiency of hyperbolic black holesη=0.56704 is lower than the Carnot efficiencyηC=0.66677.

4. Conclusion and discussion

We considered a four-dimensional charged hyperbolic black hole as a working matter to establish a black hole holographic heat engine,and use the rectangular cycle to obtain the heat engine efficiency. The efficiency is consistent with the results in the original literature.[8]Furthermore, we compared the result with the Carnot efficiency and found that when the change in entropy vanished,the charged hyperbolic black hole heat engine becomes a reversible heat engine whose efficiency is the Carnot efficiency. In addition, we also studied the efficiency of the flat case and the spherical case. The results show that the efficiency in the spherical case is higher than the flat case and the hyperbolic case. We also found that less charge corresponds to higher efficiency in the case of ?q>0. At last,when calculating in benchmark scheme, we find that the efficiency of charged hyperbolic black holes is always lower than the Carnot efficiency.

Acknowledgement

We would like to thank Wen-He Cai for helpful discussions.