Black Hole Phase Research Articles

Notes on complexity growth rate, grand potential and partition function

We examine the complexity/volume conjecture and further investigate the possible connections between complexity and partition function. The complexity/volume 2.0 states that the complexity growth rate $\mathcal{\dot{C}}\sim PV$. In the standard statistics, there is a fundamental relation among $PV$, the grand potential $\Omega$ and the partition function $\mathcal{Z}$. By using this relation, we are able to construct an ansatz between complexity and partition function. The complexity/partition function relation is then utilized to study the complexity of the thermofield double state of extended SYK models for various conditions. The relation between complexity growth rate and black hole phase transition is also discussed.

Kinetics of a phase transition for a Kerr-AdS black hole on the free-energy landscape

By treating the order parameter as a stochastic thermal fluctuating variable for small-large black hole phase transition, we investigate the kinetic process of phase transition for the Kerr-AdS (anti--de Sitter) black holes on free energy landscape. We find that the extremal points of the off shell Gibbs free energy correspond to physical black holes. For small-large black hole phase transition, the off shell Gibbs free energy exhibits a double well behavior with the same depth. Contrary to previous research for the kinetics of phase transition for the Kerr-Newman-AdS family black holes on a free energy landscape, we find that there is a lower bound for the order parameter and the lower bound corresponds to extremal black holes. In particular, the off shell Gibbs free energy is zero instead of divergent as previous work suggested for vanishing black hole horizon radius, which corresponds to the Gibbs free energy of a thermal AdS space. The investigation for the evolution of the probability distribution for the phase transition indicates that the initial stable small (large) black hole tends to switch to stable large (small) black hole. Increasing the temperature along the coexistence curve, the switching process becomes faster and the probability distribution reaches the final stationary Boltzmann distribution at a shorter time. The distribution of the first passage time indicates the time scale of the small-large black hole phase transition, and the peak of the distribution becomes sharper and shifts to the left with the increase of temperature along the coexistence curve. This suggests that a considerable first passage process occurs at a shorter time for higher temperature. The investigation of the kinetics of phase transition might provide us new insight into the underlying microscopic interactions.

Thermodynamics of 4D dS/AdS Gauss–Bonnet black holes according to consistent gravity theory in the presence of a cloud of strings

By looking at the Lovelock theorem one can infer that the gravity model given by [1] cannot be applicable for all types of 4D Einstein–Gauss–Bonnet (EGB) curved space-time. The reason for this is that in 4D space-time, the Gauss–Bonnet invariant is a total derivative and hence it does not contribute to gravitational dynamics. Hence, the authors of [2] presented an alternative consistent EGB gravity model instead of [1] by applying a break-of-diffeomorphism property. In this work, we use the alternative model to produce a de Sitter (dS)/Anti-de Sitter (AdS) black hole metric and then investigate its thermodynamic behavior in the presence of a cloud of Nambu–Goto strings. Mathematical derivations show that the resulting diagrams of pressure vs specific volume at a constant temperature are similar to that for a van der Waals gas/fluid in an ordinary thermodynamic system in the dS sector but not in the AdS background. From this, we infer that the black hole participates in the small-to-large black hole phase transition in the dS background, while it exhibits a Hawking–Page phase transition in the AdS background. In the latter case, an evaporating black hole eventually reaches an AdS vacuum space because of its instability.

Dynamic property of phase transition for non-linear charged anti-de Sitter black holes * *Supported by the National Natural Science Foundation of China (11705106, 11475107, 12075143), the Natural Science Foundation of Shanxi Province, China (201901D111315), the Natural Science Foundation for Young Scientists of Shanxi Province, China (201901D211441), the Scientific Innovation Foundation of the Higher Education Institutions of Shanxi Province (2020L0471, 2020L0472,

Understanding the thermodynamic phase transition of black holes can provide deep insights into the fundamental properties of black hole gravity and help to establish quantum gravity. In this work, we investigate the phase transition and its dynamics for the charged EPYM AdS black hole. Through reconstructing Maxwell's equal-area law, we find there exists a high-/low-potential black hole (HPBH/LPBL) phase transition, not only the pure large/small black hole phase transition. The Gibbs free energy landscape ( ) is treated as a function of the black hole horizon, which is the order parameter of the phase transition due to thermal fluctuation. From the viewpoint of , the stable HPBH/LPBL states correspond to two wells of , which have the same depth. The unstable intermediate-potential black hole state corresponds to the local maximum of . Then we focus on the probability evolution governed by the Fokker–Planck equation. Through solving the Fokker–Planck equation with different reflection/absorption boundary conditions and initial conditions, the dynamics of switching between the coexistent HPBH and LPBL phases is probed within the first passage time. Furthermore, the effect of temperature on the dynamic properties of the phase transition is also investigated.

Nonlinear-Maxwell-Yukawa de-Sitter black hole thermodynamics in a cavity: II - Grand canonical ensemble

Considering the grand canonical ensemble, we first discuss the thermodynamics of a generalized nonlinearly-charged-dS black hole enclosed in a finite spherical cavity. Such consideration means that the temperature and the potential on the wall of the cavity are fixed. Afterward, we focus on the Maxwell-Yukawa-dS black holes as a special case of investigation. The complete phase structure and stability are probed within the grand potential and the heat capacity. Our study reveals that the electric potential plays a key role in the phase portrait. Indeed, in the small values of the electric potential, the system undergoes a Hawking-Page transition and we found both phases the small and large branches. While, when the electric potential becomes relevant the large black hole phase disappears. In the end, we briefly discuss the relevant case associated with the extremal black hole one.

Thermodynamic stability and P\u2013V criticality of nonsingular-AdS black holes endowed with clouds of strings

We investigate the extended phase space thermodynamics of nonsingular-AdS black holes minimally coupled to clouds of strings in which we consider the cosmological constant (Lambda ) as the pressure (P) of the black holes and its conjugate variable thermodynamical volume (V) of the black holes. Owing to the background clouds of strings parameter (a), we analyse the Hawking temperature, entropy and specific heat on horizon radius for fixed-parameter k. We find that the strings clouds background does not alter small/large black hole (SBH/LBH) phase transition but occurs at a larger horizon radius, and two second-order phase transitions occur at a smaller horizon radius. Indeed, the G–T plots exhibit a swallowtail below the critical pressure, implying that the first-order phase transition is analogous to the liquid–gas phase transition at a lower temperature and lower critical pressure. To further examine the analogy between nonsingular-AdS black holes and a liquid–gas system, we derive the exact critical points and probe the effects of a cloud of strings on P-V criticality to find that the isotherms undergo liquid–gas like phase transition for {tilde{T}},<,{tilde{T}}_c at lower {tilde{T}}_c. We have also calculated the critical exponents identical with Van der Walls fluid, i.e., same as those obtained before for arbitrary other AdS black holes, which implies that the background clouds of strings do not change the critical exponents.

Energy and entropy compensation, phase transition and kinetics of four dimensional charged Gauss-Bonnet Anti-de Sitter black holes on the underlying free energy landscape

By treating black hole as a state in canonical ensemble or in grand canonical ensemble, we study the phase transition and the kinetics of the four dimensional charged Anti-de Sitter (AdS) black hole in Gauss-Bonnet (GB) gravity based on the free energy landscape. Below the critical temperature, the free energy landscape topography has the shape of double basins with each representing one stable/unstable black hole phase (the small or the large black hole). The thermodynamic small/large black hole phase transition is determined by the equal depths of the basins. We also demonstrate the underlying kinetics of the phase transition by studying the time evolution of the probability distribution of the state in the ensemble as well as the mean first passage time (MFPT) and the kinetic fluctuations of the state switching process caused by the thermal fluctuations. The results show that the final distribution is determined by the Boltzmann law and the MFPT and its fluctuation are closely related to the free energy landscape topography through the barrier heights between the small and the large black hole states and the ensemble temperature. Furthermore, we provide a complete description of the kinetics of phase transition by investigating the temperature dependence of the MFPT and the kinetic fluctuations with different physical parameters. It is shown that the free energy is the result of the delicate balance and competition between the two relatively large numbers, the energy and entropy multiplied by temperature compared to kT. Low energy (mass) and low entropy can give rise to a stable thermodynamic state in terms of free energy minimum (energy/mass preferred) while the high energy (mass) and high entropy (entropy preferred) can also give rise to a stable state in terms of free energy minimum. The comparable free energy barrier with respect to kT makes it possible for the switching from small size black hole state to the large size black hole state under thermal fluctuations and vice versa. When the GB coupling coupling constant increases, or the electric charge (potential) increases, or the pressure (absolute value of cosmological constant) decreases, it is easier for the small black hole state to escape to the large black hole state. Meanwhile, the inverse process becomes harder, i.e. the small (large) black hole state becomes less (more) stable.

Dynamic properties of thermodynamic phase transition for five-dimensional neutral Gauss-Bonnet AdS black hole on free energy landscape

Understanding the dynamic process of the thermodynamic phase transition can provide the deep insight into the black hole microscopic properties and structures. We in this paper study the dynamic properties of the stable small-large black hole phase transition for the five-dimensional neutral Gauss-Bonnet AdS black hole. Firstly, by using the first law of black holes, we prove that the extremal points of the free energy on the landscape denote the real black hole solutions satisfying the field equations. The local maximal and minimal points correspond to local unstable and stable black hole states, respectively. Especially, on the free energy landscape, the wells of the coexistence small and large black holes have the same depth. Then we investigate the probability evolution governed by the Fokker-Planck equation. Due to the thermal fluctuation, we find that the small (large) black hole state can transit to the large (small) black hole state. Furthermore, the first passage time is calculated. For each temperature, a single peak is presented, which suggests that there is a considerable fraction of the first passage events taking place at short time. And the higher the temperature is, the faster decrease of the probability is. These results will uncover some intriguing dynamic properties of the stable small-large black hole phase transition in modified gravity.

Magnetized Einstein\u2013Maxwell-dilaton model under an external electric field

We employ an analytic solution of a magnetized Einstein–Maxwell-dilaton gravity model whose parameters have been determined so that its holographic dual has the most similarity to the confining QCD-like theories. Analyzing the total potential of a quark–antiquark pair, we are able to investigate the effect of an electric field on different phases of the background which are the thermal AdS and black hole phases. This is helpful for better understanding of the confining character and the phases of the system. We find out that the field theory dual to the black hole solution is always deconfined, as expected. However, although the thermal AdS phase generally describes the confining phase, for quark pairs parallel to B (longitudinal case) with B>B_{mathrm {critical}} the response of the system mimics the deconfinement, since there is no IR wall in the bulk and the critical field E_s=0, as is the case for the deconfined phase. We moreover observe that in the black hole phase with sufficiently small values of mu and in the thermal AdS phase, for both longitudinal and transverse cases, the magnetic field enhances the Schwinger effect, which can be termed as the inverse magnetic catalysis (IMC). This is deduced both from the decrease of critical electric fields and decreasing the height and width of the total potential barrier the quarks are facing with. However, by increasing mu to higher values, IMC turns into magnetic catalysis, as also observed from the diagram of the Hawking–Page phase transition temperature versus B for the background geometry.

Free energy landscape and kinetics of phase transition in two coupled SYK models and the corresponding wormhole-two black hole switching

We propose that the thermodynamics and the kinetics of the phase transition between wormhole and two black hole described by the two coupled SYK model can be investigated in terms of the stochastic dynamics on the underlying free energy landscape. We assume that the phase transition is a stochastic process under the thermal fluctuations. By quantifying the underlying free energy landscape, we study the phase diagram, the kinetic time and its fluctuations in details, which reveal the underlying thermodynamics and kinetics. It is shown that the first order phase transition between wormhole and two black hole described by two coupled SYK model is analogous to the Van der Waals phase transition. Therefore, the emergence of wormhole and two black hole phases, the phase transition and associated kinetics can be quantitatively addressed in our free energy landscape and kinetic framework through the dependence on the barrier height and the temperature.

Phase transitions in 4D Gauss–Bonnet–de Sitter black holes

We investigate thermodynamic aspects of black holes in the recently formulated four dimensional Gauss-Bonnet theory of gravity, focusing on its asymptotically de Sitter ($\Lambda>0$) solutions. We take a Euclidean path integral approach, where thermodynamic data is fixed at a finite radius `cavity' outside the black hole to achieve equilibrium in the presence of the cosmological horizon. Working in the extended phase space where the cosmological constant is treated as a thermodynamic pressure, we study the phase structure of both uncharged and charged solutions, uncovering a wealth of phenomena. In the uncharged case, black holes are found to undergo either the standard Hawking-Page-like transition to empty de Sitter space or a small-large transition (akin to those seen in charged AdS black holes in pure Einstein gravity) depending on the pressure. We also find a triple point where the radiation, small, and large black hole phases coexist, and a zeroth-order phase transition within a narrow range of Gauss-Bonnet coupling parameter. Reentrant phase transitions between radiation and a small black hole also exist inside a certain parameter range. Similar phenomena are found in the charged case, with the charge parameter playing a role analogous to the Gauss-Bonnet coupling parameter in determining the phase structure.

Two approaches to quantum gravity and M-(atrix) theory at large number of dimensions

A Gaussian approximation to the bosonic part of M-(atrix) theory with mass deformation is considered at large values of the dimension [Formula: see text]. From the perspective of the gauge/gravity duality this action reproduces with great accuracy the stringy Hagedorn phase transition from a confinement (black string) phase to a deconfinement (black hole) phase whereas from the perspective of the matrix/geometry approach this action only captures a remnant of the geometric Yang–Mills-to-fuzzy-sphere phase where the fuzzy sphere solution is only manifested as a three-cut configuration termed the “baby fuzzy sphere” configuration. The Yang–Mills phase retains most of its characteristics with two exceptions: (i) the uniform distribution inside a solid ball suffers a crossover at very small values of the gauge coupling constant to a Wigner’s semicircle law, and (ii) the uniform distribution at small values of the temperatures is nonexistent.

Effects of dark energy on dynamic phase transition of charged AdS black holes

Searching for the effect of quintessence dark energy on the kinetics of black hole phase transition, we investigate in detail the dynamic phase transition of charged AdS black holes surrounded by quintessence in this paper. Based on the Gibbs free energy landscape, we obtain the analytic expression of the corresponding Gibbs free energy. As shown in $G_L-r_+$ curve at the phase transition temperature, there exist double wells with the same depth, providing further support on the finding in the former literature. By numerically solving the Fokker-Planck equation with both the initial condition and reflecting boundary condition imposed, we probe the probabilistic evolution of charged AdS black holes surrounded by quintessence. The peak denoting the initial black hole state gradually decreases while the other peak starts to grow from zero, approaching to be a stationary distribution in the long time limit with two peaks denoting the large and small black holes respectively. We also study the first passage process of charged AdS black holes surrounded by quintessence and discuss the relevant quantities. We resolve the Fokker-Planck equation by adding the absorbing boundary condition for the intermediate transition state. It is shown intuitively that the peaks located at the large (small) black hole decay very rapidly, irrespective of the initial black hole state. In all the procedures above, we have compared the cases with different choices of the state parameter of quintessence dark energy $\omega_q$. The larger $\omega_q$ is, the faster the initial black hole state decays, showing the effect of quintessence dark energy. To the best of our knowledge, it is the first probe on the influence of dark energy on the dynamic phase transition of charged AdS black hole.

General thermodynamic geometry approach for rotating Kerr-anti–de Sitter black holes

Combining with the small-large black hole phase transition, the thermodynamic geometry has been well applied to study the microstructure for the charged AdS black hole. In this paper, we extend the geometric approach to the rotating Kerr-AdS black hole and aim to develop a general approach for the Kerr-AdS black hole. Treating the entropy and pressure as the fluctuation coordinates, we construct the Ruppeiner geometry for the Kerr-AdS black hole by making the use of the Christodoulou-Ruffini-like squared-mass formula, which is quite different from the charged case. Employing the empirical observation of the corresponding scalar curvature, we find that, for the near-extremal Kerr-AdS black hole, the repulsive interaction dominates among its microstructure. While for far-from-extremal Kerr-AdS black hole, the attractive interaction dominates. The critical phenomenon is also observed for the scalar curvature. These results uncover the characteristic microstructure of the Kerr-AdS black hole. Such general thermodynamic geometry approach is worth generalizing to other rotating AdS black holes, and more interesting microstructure is expected to be discovered.

Probing black hole microstructure with the kinetic turnover of phase transition

By treating black hole as the macroscopic stable state on the free energy landscape, we propose that the stochastic dynamics of the black hole phase transition can be effectively described by the Langevin equation or equivalently by the Fokker-Planck equation in phase space. We demonstrate the turnover of the kinetics for the charged anti-de Sitter black hole phase transition, which shows that the mean first passage time is linear with the friction in the high damping regime and inversely proportional to the friction in the low damping regime. The fluctuations in the kinetics are shown to be large/small in the high/low damping regime and the switching behavior from the small fluctuations to the large fluctuations takes place at the kinetic turnover point. Because the friction is a reflection of the microscopic degrees of freedom acting on the order parameter of the black hole, the turnover and the corresponding fluctuations of the phase transition kinetics can be used to probe the black hole microstructure.

Kinetics and its turnover of Hawking-Page phase transition under the black hole evaporation

The thermodynamics and the kinetics of Hawking-Page phase transition were studied previously based on the free energy landscape. The anti--de Sitter black hole can evaporate if imposing the absorbing boundary conditions at infinity. We suggest that the kinetics of Hawking-Page phase transition should be governed by the reaction-diffusion equation, where the Hawking evaporation plays the role of the reaction on the background of the free-energy landscape. By calculating the mean first-passage time from the large black hole phase to the thermal gas phase, we show that the phase transition can occur more easily under the Hawking radiation. In particularly, a kinetic turnover is observed when increasing the ensemble temperature or the frictions. This kinetic turnover can be viewed as the dynamical phase transition to identify the time scale where Hawking evaporation process is comparable to Hawking-Page phase transition.

Formula omitted] AdS Einstein–Gauss–Bonnet black hole with Yang–Mills field and its thermodynamics

We derive an exact black hole solution for the Einstein–Gauss–Bonnet gravity with Yang–Mills field in 4D AdS spacetime and investigate its thermodynamic properties to calculate exact expressions for the black hole mass, temperature, entropy and heat capacity. The thermodynamic quantities get modification in the presence of Yang–Mills field, however, entropy remains unaffected by the Yang–Mills charge. The solution exhibits P−v criticality and belongs to the universality class of Van der Waals fluid. We study the effect of Gauss–Bonnet coupling and Yang–Mills charge on the critical behaviour and black hole phase transition. We observe that the values of critical exponents increase with the Yang–Mills charge and decrease with the Gauss–Bonnet coupling constant.

Van der Waals fluid and charged AdS black hole in the Landau theory

By introducing the general construction of Landau functional of the van der Waals system and charged AdS black hole system, we have preliminarily realized the Landau continuous phase transition theory in black hole thermodynamics. The results show that the Landau functional constructed in present paper can directly reflect the physical process of black hole phase transition. Specifically, the splitting of the global minimum of the Landau functional corresponds to the second-order phase transition of the black hole, and the transformation of the global minimum reflects the first-order phase transition of the black hole.

Thermodynamics and phase transitions of black holes in contact with a gravitating heat bath

We study the thermodynamics of a shell of self-gravitating radiation, bounded by two spherical surfaces. This system provides a consistent model for a gravitating thermal reservoir for different solutions to vacuum Einstein equations in the shell’s interior. The latter include black holes and flat space, hence, this model allows for the study of black hole phase transitions. Following the analysis of Anastopoulos C and Savvidou N (2012 Class. Quantum Grav. 29 025004), we show that the inclusion of appropriate entropy terms to the spacetime boundaries (including the Bekenstein–Hawking entropy for black hole horizons) leads to a consistent thermodynamic description. The system is characterized by four phases, two black hole phases distinguished by the size of the horizon, a flat space phase and one phase that describes naked singularities. We undertake a detailed analysis of black-hole phase transitions, the non-concave entropy function, the properties of temperature at infinity, and system’s heat capacity.

Effective thermodynamical system of Schwarzschild–de Sitter black holes from Rényi statistics

It has been known that the Schwarzschild-de Sitter (Sch-dS) black hole may not be in thermal equilibrium and also be found to be thermodynamically unstable in the standard black hole thermodynamics. In the present work, we investigate the possibility to realize the thermodynamical stability of the Sch-dS black hole as an effective system by using the R\'{e}nyi statistics, which includes the non-extensive nature of black holes. Our results indicate that the non-extensivity allows the black hole to be thermodynamically stable which gives rise to the lower bound on the non-extensive parameter. By comparing the results to ones in the separated system approach, we find that the effective temperature is always smaller than the black hole horizon temperature and the thermodynamically stable black hole in effective approach is always larger than one in separated approach at a certain temperature. There exists only the zeroth-order phase transition from the the hot gas phase to the black hole phase for the effective system while it is possible to have the transition of both the zeroth order and the first order for the separated system.