Effects of dark energy on P-V criticality of charged AdS black holes

In this Letter, we investigate the effects of dark energy on $P-V$ criticality of charged AdS black holes by considering the case of the RN-AdS black holes surrounded by quintessence. By treating the cosmological constant as thermodynamic pressure, we study its thermodynamics in the extended phase space. It is shown that quintessence dark energy does not affect the existence of small/large black hole phase transition. For the case $\omega_q=-2/3$ we derive analytic expressions of critical physical quantities, while for cases $\omega_q\neq-2/3$ we appeal to numerical method for help. It is shown that quintessence dark energy affects the critical physical quantities near the critical point. Critical exponents are also calculated. They are exactly the same as those obtained before for arbitrary other AdS black holes, which implies that quintessence dark energy does not change the critical exponents.Comment: 13 pages, 2 figure

Holographic Heat engine within the framework of massive gravity

Heat engine models are constructed within the framework of massive gravity in this paper. For the four-dimensional charged black holes in massive gravity, it is shown that the heat engines have a higher efficiency for the cases $m^2>0$ than for the case $m=0$ when $c_1<0, c_2<0$. Considering a specific example, we show that the maximum efficiency can reach $0.9219$ while the efficiency for $m=0$ reads $0.5014$. The existence of graviton mass improves the heat engine efficiency significantly. The situation is more complicated for the five-dimensional neutral black holes. Not only the $c_1, c_2, m^2$ exert influence on the efficiency, but also the constant $c_3$ corresponding to the third massive potential contributes to the efficiency. When $c_1<0, c_2<0, c_30$ is higher than that of the case $m=0$. By studying the ratio $\eta/\eta_C$, we also probe how the massive gravity influences the behavior of the heat engine efficiency approaching the Carnot efficiency.Comment: 9pages,4figure

Ratio of critical quantities related to Hawking temperature-entanglement entropy criticality

We revisit the Hawking temperature$-$entanglement entropy criticality of the $d$-dimensional charged AdS black hole with our attention concentrated on the ratio $\frac{T_c \delta S_c}{Q_c}$. Comparing the results of this paper with those of the ratio $\frac{T_c S_c}{Q_c}$, one can find both the similarities and differences. These two ratios are independent of the characteristic length scale $l$ and dependent on the dimension $d$. These similarities further enhance the relation between the entanglement entropy and the Bekenstein-Hawking entropy. However, the ratio $\frac{T_c \delta S_c}{Q_c}$ also relies on the size of the spherical entangling region. Moreover, these two ratios take different values even under the same choices of parameters. The differences between these two ratios can be attributed to the peculiar property of the entanglement entropy since the research in this paper is far from the regime where the behavior of the entanglement entropy is dominated by the thermal entropy.Comment: Comments welcome. 11 pages, 3 figure

Heat engine in the three-dimensional spacetime

We define a kind of heat engine via three-dimensional charged BTZ black holes. This case is quite subtle and needs to be more careful. The heat flow along the isochores does not equal to zero since the specific heat $C_V\neq0$ and this point completely differs from the cases discussed before whose isochores and adiabats are identical. So one cannot simply apply the paradigm in the former literatures. However, if one introduces a new thermodynamic parameter associated with the renormalization length scale, the above problem can be solved. We obtain the analytical efficiency expression of the three-dimensional charged BTZ black hole heat engine for two different schemes. Moreover, we double check with the exact formula. Our result presents the first specific example for the sound correctness of the exact efficiency formula. We argue that the three-dimensional charged BTZ black hole can be viewed as a toy model for further investigation of holographic heat engine. Furthermore, we compare our result with that of the Carnot cycle and extend the former result to three-dimensional spacetime. In this sense, the result in this paper would be complementary to those obtained in four-dimensional spacetime or ever higher. Last but not the least, the heat engine efficiency discussed in this paper may serve as a criterion to discriminate the two thermodynamic approaches introduced in Ref.[29] and our result seems to support the approach which introduces a new thermodynamic parameter $R=r_0$.Comment: Revised version. Discussions adde