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Thermodynamics and overcharging problem in extended phase space of charged AdS black holes with cloud of strings and quintessence under charged particle absorption

  • The thermodynamics and overcharging problem in RN-AdS black holes with a cloud of strings and quintessence are investigated by the absorption of scalar particles and fermions in extended phase space. The cosmological constant is treated as the pressure of the black hole. The parameters related to quintessence and the cloud of strings are treated as thermodynamic variables. We find that the first law of thermodynamics is satisfied and the second law of thermodynamics is indefinite. Furthermore, we find that near-extremal and extremal black holes cannot be overcharged.
  • Statistical equilibrium is a basic assumption in numerous theoretical studies [1-5]. In these studies, the primary focus has been on phase transition [1, 2] and nuclear multifragmentation [3-5]. Many experimental studies have been devoted to testing this theoretical assumption [6-12]. These studies were motivated by the strong agreements obtained between various observable trends related to the asymptotically resulting fragments and a variety of multifragmentation models. In the frameworks of these multifragmentation models, the freezing out process is very important. For example, both statistical and chemical equilibrium are assumed between the produced fragments in a statistical multifragmentation model (SMM) [3]. A hot source with mass and charge (A0, Z0) at temperature T expands to a freeze-out volume. Fragments are not allowed to overlap one another, and they are placed into a volume V (freeze-out volume). The source size, its excitation energy, and its volume are the basic quantities of the statistical models. To obtain these data, indirect evaluations must be carried out via comparisons between the experimental data and statistical model predictions. However, the correlation between the experimental data and bulk properties is far from simple. When the system reaches the freeze-out stage, the primary fragments may be excited. Understanding the multifragmentation phenomenon is difficult owing to the decay of the primary fragments. The detected fragments are cold remnant fragments. To facilitate a comparison with the experimental data, the SMM requires not only the information of the excited sources but also the information of the primary fragments. In fact, different primary configurations may lead to the same final results because of the compensatory effects between the primary and secondary emission mechanisms [13]. The difficulty of determining the freeze-out volume is reflected in the varying values. Different freeze-out volumes have been obtained in many studies, ranging from 2.5V0 to 9V0 [14-16], where V0 is the volume corresponding to normal nuclear matter density.

    Heavy-ion collisions are the only means of studying the properties of hot nuclei [17]. In such collisions, the dynamical process can be divided into three stages. (i) The system driven by intensive interactions between nucleons evolves toward thermalization, and fast particles leave the system. The time interval of this stage is approximately several tens of fm/c. The particle emission of this stage is pre-equilibrium emission. (ii) The hot nuclear residue expands and breaks up into hot primary fragments. The produced fragments are in the freeze-out stage. (iii) The primary fragments are de-excited by emitting particles and gamma rays to the final ground states.

    In the present work, the focus is on the freeze-out volume. Experimental studies are extremely valuable for determining the freeze-out volume. The extraction of the volume from the measured yields of particles is discussed, and the temperature and density are studied using the yields and quantum fluctuations of light charged particles (Z 2 LCP) [18]. However, the sources of LCPs are complex. The LCPs, which are measured experimentally, may originate from several sources: (i) pre-equilibrium emission, (ii) the composite excited system at the freeze-out stage, and (iii) sequential decay of excited fragments. LCPs cannot originate from an approximate single system. Therefore, the effects of pre-equilibrium emission and sequential decay on the determination of the freeze-out volume are studied in this work.

    In this work, an attempt is made to study the influence of pre-equilibrium emission and secondary decay on the determination of the freeze-out volume via the isospin-dependent quantum molecular-dynamics (IQMD) model incorporating the statistical decay model GEMINI [19-21]. In this model, each nucleon is represented by a coherent state of a Gaussian wave packet

    ϕi(ri,t)=1(2πL)3/4e[riri0(t)]24Leiripi0(t),

    (1)

    where ri0 and pi0 are the average values of the position and momentum of the ith nucleon, respectively, and L is related to the extension of the wave packet. L is equal to σ2r, where σr is the width of the wave packet. The width of the wave packet affects the stability of the nuclei at their ground state and the charge distribution of fragments in heavy-ion collisions. If the width of the wave packet is smaller than 1 fm, the “spurious” emission number of nucleons increases sharply with the decrease in the width of the wave packet. An excessively large wave-packet width causes the central densities to be obviously higher than the normal density [22]. When the width of the wave packet is 1.1 fm, the stable nucleus can be produced. The corresponding value of L is 1.21 fm2. The total N-body wave function is assumed to be the direct product of these coherent states. Through a Wigner transformation of the wave function, the one-body phase-space distribution function for N-distinguishable particles is given by

    f(r,p,t)=ni=11(π)3e[rri0(t)]22Le[ppi0(t)]22L2.

    (2)

    The time evolutions of the nucleons in the system under the self-consistently generated mean field are governed by Hamiltonian equations of motion

    ˙ri0=pi0H,˙pi0=ri0H,

    (3)

    where the Hamiltonian H is expressed as

    H=Ekin+UCoul+V(ρ)dr.

    (4)

    In the above, the first term Ekin is the kinetic energy, the second term UCoul is the Coulomb potential energy, and the third term is the local nuclear potential energy. Each term of the local potential energy-density function V(ρ) in this work is

    Vsky=α2ρ2ρ0+βγ+1ργ+1ργ0,

    (5)

    Vsur=gsur2(ρ)2ρ0+gisosur2(ρnρp)2ρ0,

    (6)

    Vmdi=gτρ8/3ρ5/30.

    (7)

    In this case, Vsky, which includes the two-body and three-body interaction terms, describes the saturation properties of nuclear matter. Vsur is the surface term that describes the surface of finite nuclei. Vmdi is the momentum-dependent interaction term. The symmetry potential energy-density functional Vsym is

    Vsym=Csym2(ρnρp)2ρ0.

    (8)

    The parameters used in this study are α = −168.40 MeV, β = 115.90 MeV, γ = 1.50, gsur = 92.13 MeV fm2, gisosur = -6.97 MeV fm2, Csym = 38.13 MeV, and gτ = 0.40 MeV. The corresponding compressibility is 271 MeV [23]. The fragments are identified by a minimum spanning-tree algorithm. The nucleons with a relative distance of R0 3.5 fm and momentum of P0 250 MeV/c belong to a fragment.

    In this study, the dynamical description is used not only for the excitation stage but also for intermediate-mass-fragment (IMF) emission. Following the excitation stage, the time evolution in the IQMD code continues until the excitation energy of the heaviest hot fragment decreases to a certain value Estop in each event. If the excitation energy is lower than Estop [21], the IQMD calculation stops and the charge, mass, excitation energy, and momentum of each hot fragment are recorded. The outputs of the IQMD code are the hot fragments. To obtain the cold fragments, emission of light particles from the hot fragments is achieved using the statistical code GEMINI.

    To study the freeze-out volume, the central collisions of small-mass projectiles and large-mass targets are used to produce hot nuclei. For such reaction systems, sufficient nucleons exist in the overlap volume to experience the required collisions for hot-nuclei thermalization [24]. To reduce the effects of the mass range of the hot nuclei on the proton production, the narrow mass number range of the hot nuclei is required to be 190 A 200. The selection method of the hot nuclei is the same as that presented in Ref. [25]. It is worth noting that the use of hot nuclei with a mass number range of 190 A 200 only satisfies the requirement of the event number. In this work, by using the reaction system 36Ar + 197Au with beam energies of 50, 60, and 70 MeV/u, the hot nuclei have a wide mass number range (approximately 160-230). If another mass number range is selected, more events need to be calculated owing to the low production of the hot nuclei.

    Using the hot nuclei, the freeze-out temperatures can be calculated by the isotope-yield-ratio method of Albergo et al. [26, 27]. The corresponding expression is

    TBeLi=11.3MeV/ln(1.8Y9Be/Y8LiY7Be/Y6Li).

    (9)

    Using Eq. (9), only the apparent temperature (Tapp) can be studied. Cold fragments are used to calculate Tapp. However, the primary fragments are normally excited at the freeze-out stage. To calculate the freeze-out temperature (T0), one can connect T0 and Tapp by the linear approximation T0 = 1.2Tapp [8].

    At the freeze-out stage, protons (p), neutrons (n), tritium, etc., follow Fermi statistics, whereas deuterium, α, etc., should follow Bose statistics. The temperature and density of nuclear systems have been studied by employing distributions of particles [28, 29]. In this work, only protons that are abundantly produced in the collisions are studied. In the freeze-out stage, the density of the protons can be determined via the Fermi distribution

    ρFp=4π(2m)3/2h30ε1/2dεeεμT+1,

    (10)

    where m is the mass of the protons.

    The multiplicity for a proton can be expressed as

    N=4πV(2m)3/2h30ε1/2dεeεμT+1,

    (11)

    (ΔN)2=T(Nμ)T,V.

    (12)

    Substituting Eq. (11) into Eq. (12), the following is obtained:

    (ΔN)2=4πV(2m)3/2h30eεμTε12dε(eεμT+1)2.

    (13)

    The multiplicity fluctuation for a proton (MFp) can be determined by [30]

    (N)2N=0ε12dεeεμT(eεμT+1)20ε12dε1eεμT+1.

    (14)

    The MFp values can be calculated by the IQMD code or measured experimentally. Using the MFp values, the integral variable μ can be solved numerically by Eq. (14). The freeze-out temperature can be calculated by Eq. (9). Substituting μ and the freeze-out temperature into Eq. (10) yields the density of the protons at freeze-out. The freeze-out volume V can be calculated by N/ρFp, where N is the proton average multiplicity at freeze-out.

    To calculate the freeze-out volume, the effects of pre-equilibrium emission and sequential decay should be bypassed. To define the equilibrium and freeze-out moment, the time evolution of the quadrupole momentum and IMFs, i.e., fragments with Z 3 are depicted in Figs. 1(a) and 1(b), respectively. These are calculated for the system 36Ar + 197Au at a 50 MeV/u bombarding energy and center collisions. The quadrupole moment of the momentum distribution is determined by

    Figure 1

    Figure 1.  (a) Time evolution of quadrupole momentum for maximum mass cluster and (b) IMF multiplicity for reaction system.

    Qp=(2p2zp2xp2y)f(r,p,t)drdp.

    (15)

    The quadrupole moment of the momentum is calculated by all nucleons that belong to the largest cluster in the center of mass of the largest cluster.

    It can be observed from Fig. 1(a) that the quadrupole increases rapidly at 10 fm/c. At this moment, the projectile and target are in contact with one another. At approximately 80 fm/c, the quadrupole recovers to zero again. The momentum of nucleons reaches an isotropic distribution at 100 fm/c [24]. Thus, the protons emitted before 100 fm/c comprise pre-equilibrium emission. With the change in the reaction time, the hot nuclei expand and break into primary fragments. The hot nuclei reach the freeze-out stage, and the freeze-out moment can be estimated from the time evolution of the multiplicity of IMFs. It can be observed that the multiplicity of IMFs ends its variation at approximately 400 fm/c. The protons produced after 400 fm/c constitute secondary decay. Therefore, in this study, the protons are divided into four parts: (i) pre-equilibrium emission (PEp), (ii) protons produced in the freeze-out stage (FOp), (iii) secondary decay (SDp), and (iv) PEp+FOp+SDp (TOTAL).

    In the following discussion, the focus is on the moderate excitation energy range (6-8 MeV/nucleon). To produce moderate excitation hot nuclei, three beam energies are selected, namely 50, 60, and 70 MeV/u. The reaction system is 36Ar + 197Au. At a low excitation energy, evaporation will be the main de-excitation process. The protons originate from the surface of the hot nuclei and not from the freeze-out volume. At a moderate excitation energy (approximately 7 MeV/nucleon), an IMF has a peak value [9], and the hot nuclear system is fully broken. The nucleus breaks into pieces, with the large fragments representing the liquid and the very small ones representing the vapor. The Fermi-gas approach is well justified for this weakly interacting system. More importantly, a nuclear liquid-gas phase transition may occur at a moderate excitation energy [10, 31]. Figure 2 depicts the proton yields of different stages for different excitation energies. It can be observed that the proton multiplicity of pre-equilibrium emission is approximately 4. Most of the protons are produced at freeze-out and via the secondary decay process. The production of secondary decay is approximately 10. Thus, most of the protons are produced by the de-excitation process. Moreover, Fig. 2 indicates that the proton production exhibits a slow increase with an increasing excitation energy.

    Figure 2

    Figure 2.  (color online) Proton yield at different stages (PEp, FOp, SDp, and TOTAL) as a function of excitation energy.

    In heavy-ion collisions, the hot nuclei will be de-excited by disintegration. The de-excitation process can be light-particle (Z 2) evaporation or IMF (Z 3) emission. The competition of the two modes determines the de-excitation process. The fragment charge distribution reflects the de-excitation process of the hot nuclei. When the excitation energy is low, the charge distribution has a "U"-shaped characteristic, corresponding to the evaporation event. When the excitation energy is high, the charge distribution has a rapidly decreasing charge distribution, corresponding to the vaporization event [11]. Figure 3 presents the charge distribution as a function of the charge number Z of the fragments. The experimental data are obtained from Ref. [12]. The square symbols indicate the calculated values. The behavior of the calculated charge distribution is generally in agreement with the data. Therefore, our calculation can appropriately describe the de-excitation process of the hot nuclei. The main difference in the calculations compared with the experimental data is the overestimation of the proton yields. However, for the following discussion, the relative proton yields at different stages are more important. The main focus of this study is on the effects of the relative proton yields.

    Figure 3

    Figure 3.  (color online) Charge distribution N(Z) in central 197Au + 197Au collisions at 35 MeV/u. The experimental data are obtained from Ref. [12].

    The freeze-out temperatures are shown in Fig. 4(a). The calculation points are plotted for 0.2-MeV/nucleon-wide bins in excitation energy per nucleon. The MFp versus excitation energy per nucleon is plotted in Fig. 4(b). The proton yield of the secondary decay is approximately 3 times that of the pre-equilibrium emission (see Fig. 2). However, the normalized fluctuations are more easily affected by the pre-equilibrium emission. The secondary decay process is more complex than the pre-equilibrium emission process. Many de-excitation routes are available for secondary decay, and therefore, the competition among the different de-excitation routes increases the fluctuation of the proton production.

    Figure 4

    Figure 4.  (color online) (a) Freeze-out temperatures and (b) multiplicity fluctuation for proton vs. excitation energy per nucleon E/A.

    The freeze-out volume versus excitation energy is plotted in Fig. 5. The freeze-out volume can be calculated by four groups of protons (FOp+PEp, FOp, FOp+SDp, and TOTAL). The freeze-out volume calculated by FOp is approximately 2 times that of FOp+PEp. The freeze-out volumes are almost the same between TOTAL and FOp+ PEp. The difference in the freeze-out volume between FOp and FOp+SDp is smaller than that between FOp and FOp+PEp.

    Figure 5

    Figure 5.  (color online) Freeze-out volume calculated by protons produced at different stages as a function of excitation energy.

    The study of the freeze-out volume is helpful for gaining an improved understanding of the multifragmentation process, which may offer the possibility for investigating the nuclear liquid-gas transition. Experimental studies are indespensible to obtain the freeze-out volume and understand the freeze-out concept. However, the particles detected experimentally include the information of pre-equilibrium and secondary decay. The determination of the freeze-out information is affected by the interference of the pre-equilibrium and secondary decay. Therefore, it is necessary to study the effects of pre-equilibrium emission and secondary decay on the determination of the freeze-out information. In this work, the freeze-out volume is studied by the multiplicity fluctuation for a proton. The calculations indicate that pre-equilibrium emission and secondary decay will affect the determination of the freeze-out information. When using the quantum fluctuations of the proton to study the freeze-out volume, the effect of pre-equilibrium emission is more obvious. However, the present results calculated by the IQMD model depend on the model parameters. The use of different model parameters may lead to different results. Therefore, the effects of different model parameters on the determination of the freeze-out volume should be studied in the future.

    A study on the freeze-out volume from the yields of protons emitted in heavy-ion collisions has been presented in this paper. The study of the freeze-out volume is very important for understanding the multifragmentation process, which opens the possibility for investigating the liquid-gas coexistence region. Experimental studies are indispensable to obtain the freeze-out volume and to understand the freeze-out concept. However, the particles that are detected experimentally include the information of pre-equilibrium emission and secondary decay. The determination of the freeze-out information is affected by the interference of pre-equilibrium and secondary decay.

    Owing to the effects of pre-equilibrium emission and secondary decay, the percentage of protons in the freeze-out stage is only approximately 50%. In the de-excitation process, the proton yield produced by secondary decay is approximately 3 times that of pre-equilibrium emission. However, the normalized fluctuations are more easily affected by pre-equilibrium emission because the secondary decay process is more complex. The competition among the different de-excitation routes increases the fluctuation of the proton production. Therefore, when using the multiplicity fluctuation of protons to study the freeze-out volume, more attention should be paid to pre-equilibrium emission.

    It should be stressed that the present results are based on specific IQMD model parameters. If different model parameters are used, different results may be obtained. Therefore, these effects should be studied in the future.

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Jing Liang, Benrong Mu and Jun Tao. Thermodynamics and overcharging problem in the extended phase space of charged AdS black holes with cloud of strings and quintessence under charged particle absorption[J]. Chinese Physics C. doi: 10.1088/1674-1137/abd085
Jing Liang, Benrong Mu and Jun Tao. Thermodynamics and overcharging problem in the extended phase space of charged AdS black holes with cloud of strings and quintessence under charged particle absorption[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abd085 shu
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Thermodynamics and overcharging problem in extended phase space of charged AdS black holes with cloud of strings and quintessence under charged particle absorption

  • 1. Physics Teaching and Research section, College of Medical Technology, Chengdu University of Traditional Chinese Medicine, Chengdu 611137, China
  • 2. Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu 610064, China

Abstract: The thermodynamics and overcharging problem in RN-AdS black holes with a cloud of strings and quintessence are investigated by the absorption of scalar particles and fermions in extended phase space. The cosmological constant is treated as the pressure of the black hole. The parameters related to quintessence and the cloud of strings are treated as thermodynamic variables. We find that the first law of thermodynamics is satisfied and the second law of thermodynamics is indefinite. Furthermore, we find that near-extremal and extremal black holes cannot be overcharged.

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    I.   INTRODUCTION
    • Since Stephen Hawking proved that black holes emit quantum radiation with a temperature T=κ2π [1], it has been believed that a black hole can be treated as a thermodynamic system. Until now, there has been much work on the thermodynamics of black holes, such as the four laws of thermodynamics [2, 3], phase transitions [4] and quantum effects [5, 6]. In general, the first law of thermodynamics of a black hole is written as:

      dM=TdS+ΦdQ,

      (1)

      where M is the mass, T is the Hawking temperature of the black hole, S is the entropy, Φ is the electric potential and Q is the electric charge. The mass is usually interpreted as the enthalpy [7]. It is clear that there is no PdV term in Eq. (1), which corresponds to the change in volume under pressure P. When the cosmological constant, Λ, is treated as the pressure of the black hole [8-13] and the volume of the black hole is defined as the thermodynamic variable conjugate to the pressure [14], Eq. (1) is modified as:

      dM=TdS+VdP+ΦdQ.

      (2)

      The relations between P, Λ and V are P=Λ8π, V=(MP)S,Q. High-precision observations have recently confirmed the existence of a gravitationally repulsive interaction at a global scale (cosmic dark energy) [15]. It is found that one type of dark energy model produces some gravitational effect when it surrounds black holes. For this type of dark energy, the equation of state parameters is in the interval [1,13] [16]. This type of dark energy model is called quintessence dark energy, or quintessence for short. In this case, the first law of thermodynamics is given by [17]

      dM=TdS+VdP+ΦdQ12r3ωq+dα,

      (3)

      where α is a positive normalization factor. There has been much interest in studying the physics of black holes surrounded by quintessence [18-35].

      According to string theory, nature can be represented by a set of extended objects (such as one-dimensional strings) rather than point particles. Therefore, understanding the gravitational effects caused by a set of strings is necessary. This can be achieved by solving Einstein's equations with a finite number of strings. The results obtained by Letelier show that the existence of a cloud of strings will produce a global origin effect related to a solid deficit angle. Moreover, the solid deficit angle depends on the parameters that determine the existence of the cloud [36]. Therefore, the existence of a cloud of strings will have an impact on black holes. When we consider the existence of the cloud of strings, the first law of thermodynamics takes the form:

      dM=TdS+VdP+ΦdQr+2da,

      (4)

      where a is the state parameter of cloud of strings. The effect of the cloud of strings on black holes has been explored for various black holes [37-42]. As noted in Ref. [43], if the parameters related to the cloud of string and quintessence are considered as extensive thermodynamic parameters, the first law of thermodynamics of black holes is modified as:

      dM=TdS+VdP+ΦdQ12r3+ωqdαr+2da.

      (5)

      There has been much interest in deducing and discussing the physical properties of various black holes when they are surrounded by cloud of strings and quintessence [43-50].

      An important feature of a black hole is its horizon, no matter what matter can escape through it. There is a gravitational singularity at the center of the black hole, which is hidden by the event horizon. At the singularity, all the laws of physics break down. In order to avoid this phenomenon, Penrose proposed the weak cosmic censorship conjecture (WCCC) in 1969 [51, 52]. The WCCC claims that the singularity is always hidden by the event horizon and cannot be seen by an observer at infinite distance. Although the WCCC's correctness is widely accepted, there is no complete evidence to prove it and only its validity can be tested. Gedanken experiments are an effective method of testing the validity of the WCCC [53]. In these thought experiments, a test particle with sufficient energy, charge and angular momentum is thrown into the black hole. After the black hole absorbs the test particle, if the horizon is destroyed, the singularity becomes a naked singularity. In this case, the black hole is overcharged and the WCCC is violated. Conversely, if the horizon is not destroyed, the singularity is surrounded by the horizon. Consequently, the black hole is not overcharged and the WCCC is valid. The validity of the WCCC has been tested in this way for various black holes [54-74]. Recently, Sorce and Wald proposed a new version of the gedanken experiment to examine the WCCC for Kerr-Newman black holes [75]. The result of this experiment shows that the event horizon of a Kerr-Newman black hole cannot be destroyed. After that, using this experiment, the WCCC was found to be valid for a series of black holes [76-79]. Another way of destroying the event horizon of a black hole to test the validity of the WCCC is to use test fields to replace test particles in the gedanken experiment [80], a method which has been further developed in Refs. [81-92].

      In this paper, we investigate the thermodynamics and overcharging problem in a RN-AdS black hole with a cloud of strings and quintessence, by charged particle absorption in extended phase space. Due to the existence of the cloud of strings and quintessence, the constants related to them are also taken into account in the calculation. The structure of this paper is as follows. In Section II, we review the thermodynamics of a RN-AdS black hole with a cloud of strings and quintessence. In Section III, the absorptions of scalar particles and fermions are discussed. In Section IV, the first and second laws of thermodynamics are investigated in extended phase space. In Section V, the overcharging problem is tested in near-extremal and extremal black holes. Our results are summarized in Section VII.

    II.   BLACK HOLE SOLUTION
    • The metric of a RN-AdS black hole surrounded by a cloud of strings and quintessence in 4-dimensional space-time is given by [18, 44, 47]

      ds2=f(r)dt21f(r)dr2r2(dθ2+sin2θdϕ2),

      (6)

      with

      f(r)=1a2Mr+Q2r2αr3ωq+1Λr23.

      (7)

      In the above equation, Λ is the cosmological constant related to the AdS space radius l by Λ=3/l2, and M and Q are the mass and charge of the black hole, respectively. a is the integral constant caused by the cloud of strings and α is a normalization constant related to the quintessence, with density ρq

      ρq=α23ωqr3(ωq+1).

      (8)

      In Fig. 1, the graphs of the function f(r) are shown for different values of the parameters a, α and ωq, which represent the presence of the cloud of strings and the quintessence.

      Figure 1.  (color online) The function f(r) for different values of a, α and ωq. We choose M=1 and Q=0.8.

      For a non-extremal black hole, the equation f(r)=0 has two positive real roots, r+ and r. r+ represents the radius of the event horizon. For an extremal black hole, f(r)=0 has only one root, r+. The mass of the black hole can be represented by:

      M=12(rar+Q2r2αr3ωq+r3l2).

      (9)

      The Hawking temperature takes the form:

      T=f(r+)4π=14π(2Mr2+2Q2r3++(3ωq+1)αr3ωq+2++2r+l2).

      (10)

      The entropy and the potential of the black hole are:

      S=πr2+,

      (11)

      Φ=At(r+)=Qr+.

      (12)

      In previous studies, the cosmological constant was treated as a constant. Recently, however, the thermodynamic pressure of the black hole has been introduced into the laws of thermodynamics, and the cosmological constant treated as a variable related to pressure. The relationship between the cosmological constant and pressure is [8-13]

      P=Λ8π=38πl2.

      (13)

      The first law of thermodynamics in the extended phase space is written as:

      dM=TdS+VdP+φdQ+γdα+ϰda,

      (14)

      where

      γ=12r3ωq+,ϰ=r+2.

      (15)

      In the above equation, the volume is given by:

      V=(MP)S,Q=4πr3+3.

      (16)

      The mass of the black hole M is defined as its enthalpy. Hence, the relationship between enthalpy, internal energy and pressure is [7, 10]:

      M=U+PV.

      (17)
    III.   PARTICLE ABSORPTION

      A.   Scalar particle absorption

    • In curved space-time, the motion of a charged scalar particle satisfies the Klein-Gordon equation:

      1g(xμiqAμ)[ggμν(xνiqAν)]ΨSm22ΨS=0,

      (18)

      where ΨS is the scalar field, and m and q are the mass and charge of the particle, respectively. Using the WKB approximation [93-95], the wave function is written as

      ΨS=exp(iI+I1+O()).

      (19)

      Inserting Eq. (19) and the contravariant metric components of the 4-dimensional black hole into the Klein-Gordon equation, we obtain:

      f1(tIqAt)2f(rI)21r2(θI)21r2sin2θ(φI)2+m2=0.

      (20)

      Considering the symmetry of space-time, it is necessary to carry out the separation of variables in the action

      I=ωt+W(r)+S(θ,φ),

      (21)

      where ω is the energy of the absorbed scalar particle. Substituting the above separated action into Eq. (20) and simplifying it yields:

      rW=±(ω4qQr)2+[m21r2(θS)21r2sin2θ(φS)2]ff.

      (22)

      In the above equation, +() indicates the situation of ingoing (outgoing) particles. Since we suppose that the particles are completely absorbed by the black hole, the negative sign is ignored [96]. Defining pr=rI=rW, the above equation is modified as:

      pr =grrpr=(ω4qQr)2+[m21r2(θS)21r2sin2θ(φS)2]f.

      (23)

      Here we consider the situation of absorbed particles near the horizon, which means f(r)0. Then Eq. (23) is simplified to

      pr=ωqΦ,

      (24)

      where Φ=Qr+ is the electric potential. Equation (24) is the relationship between the momentum, the energy and the charge of the ingoing particle. When ω<qΦ, the energy of the black hole flows out from the event horizon and superradiation happens. When ω=qΦ, the energy of the black hole does not change. In this paper, it is assumed that ωqΦ, which implies superradiation does not occur. Equation (24) plays an important role in the discussion of black hole thermodynamics, and is recovered by fermion absorption in the next section.

    • B.   Fermion absorption

    • In curved space-time, the motion of a charged fermion obeys the Dirac equation:

      iγμ(μ+ΩμiqAμ)ΨF+m0ΨF=0,

      (25)

      where m and q are the mass and charge of the fermion, respectively. Ωμi2ωμabΣab, ωabμ is the spin connection defined by the normal connection and the tetragonal eλb. The relation between the spin connection, the normal connection and the tetragonal is

      ωμab=eνaeλbΓνμλeλbμeλa.

      (26)

      The Greek index rises and falls with the curved metric gμν. The Latin index is dominated by the flat metric ηab. To construct the tetrad, the following definition is needed:

      gμν = eμaeνbηab,ηab = gμνeμaeνb,eμaeνa = δμν,eμaeμb = δba.

      (27)

      The Lorentz spinor generators are defined by:

      Σab=i4[γa,γb],{γμ,γν}=2ηab.

      (28)

      Then the γμ is constructed in curved space-time as:

      γμ=eμaγa,{γμ,γν}=2gμν.

      (29)

      For a fermion with a spin of 1/2, its wave function must be described as both spin-up and spin-down. We first describe the spin-up wave function. The wave function takes the form:

      ΨF=(A0B0)exp(iI(t,r,θ,φ)).

      (30)

      where A, B and I are functions of t, r, θ, ϕ. For the metric (6), we choose:

      eμa=diag(f,1/f,r,rsinθ).

      (31)

      Then the γμ matrices are written as:

      γt=1f(r)(i00i),γθ=r(0σ1σ10),γr=f(r)(0σ3σ30),γφ=rsinθ(0σ2σ20),

      (32)

      where σi are the Pauli matrices, which are given by:

      σ1=(0110),σ2=(0ii0),σ3=(1001).

      (33)

      Inserting the spin connection, wave function and gamma matrices into the Dirac equation, we obtain:

      iA1f(tIqAt)BgrI+Am0=0,

      (34)

      iB1f(tIqAt)AgrI+Bm0=0,

      (35)

      A[rθI+irsinθφI]=0,

      (36)

      B[rθI+irsinθφI]=0.

      (37)

      Equations (36) and (37) are simplified to one equation and yield r2(θI)2+r2sin2θ(φI)2=0. In previous studies, the contribution of the angular part did not affect the results of tunneling radiation when the quantum gravity effects were not considered [97]. Since the radial action is determined by the first two of the above four equations, we mainly focus on them. To solve the question we are addressing, it is necessary to use the separation of variables:

      I=ωt+W(r)+Θ(θ,φ).

      (38)

      In the above equation, ω is the energy of the ingoing fermion. Inserting Eq. (38) into Eqs. (34) and (35) yields:

      f2(rW)2(ω4qQr)2m20f=0.

      (39)

      Simplifying Eq. (39), we have:

      rW=±(ω4qQr)2+m20ff,

      (40)

      where the +/ corresponds to the cases of the ingoing/outgoing fermion. In the work of Gwak [96], the positive sign in the above equation was selected. Thus, we get

      pr=grrpr=(ω4qQr)2+m20f.

      (41)

      Near the event horizon, f0. The above equation is modified as

      pr=ωqΦ,

      (42)

      Therefore, Eq. (24) can be recovered by fermion absorption. The above discussion calculates the spin-up state. For the spin-down state, the result is the same as the spin-up state. It is clear that the results obtained by scalar particle absorption and fermion absorption are the same.

    IV.   THERMODYNAMICS AND PARTICLE ABSORPTION WITH CONTRIBUTION OF PRESSURE AND VOLUME
    • Recently, an interesting method has been adopted to treat the cosmological constant as a thermodynamic variable. From this point of view, the cosmological constant is not a fixed value but a thermodynamic variable, and represents quite consistent behaviors with other thermodynamic variables [8, 14]. In this extended thermodynamics, the cosmological constant plays the role of pressure P

      P=Λ8π=38πl2.

      (43)

      In this case, the mass of the black hole is no longer equivalent to the internal energy, but is seen as a gravitational version of the chemical enthalpy. The thermodynamic volume of the black hole is:

      V=MP=4πr3+3.

      (44)

      In order to satisfy the Smarr relation, we need to further expand the phase space by elevating any dimensional parameter of the theory to a thermodynamic variable and introducing the associated conjugate [98, 99]. These parameters and their conjugations add additional terms to the first law of thermodynamics. In particular, the parameters related to the cloud of strings and quintessence are considered to be thermodynamic phase space variables, and the associated conjugates are:

      γ=Mα=12r3ωq+,ϰ=Ma=r+2.

      (45)

      The quintessence parameters γ and α are playing the same role as pressure and volume in the first law to make it consistent with the Smarr relation. The same is true for the parameters related to the cloud of strings, i.e. ϰ and a. As shown in Ref. [44], for a 4-dimensional RN-AdS black hole surrounded by a cloud of strings and quintessence, all these quantities satisfy the following generalized Smarr formula:

      M=2TS2VP+ΦQ+(3ωq+1)γα.

      (46)

      When the black hole absorbs a particle, the changes in the internal energy and charge of the black hole are equal to the energy and charge of the particle, which take the form:

      ω=dU=d(MPV), q=dQ,

      (47)

      where MPV is the internal energy of the black hole in the extended phase space. Therefore, Eq. (42) is written as:

      dU=Qr+dQ+pr.

      (48)

      Thus, after a charged particle with energy ω and charge q enters the black hole horizon, the black hole changes from the initial state (M,Q,P,a,α,r+) to the final state (M+dM,Q+dQ,P+dP,a+da,α+dα,r++dr+). The functions f(M,Q,P,a,α,r+) and f(M+dM,Q+dQ,P+dP,a+da,α+dα,r++dr+) satisfy

      f(M,Q,P,a,α,r+)= f(M+dM,Q+dQ,P+dP,a+da,α+dα,r++dr+)=0.

      (49)

      The relation between the functions f(M,Q,P,a,α,r+) and f(M+dM,Q+dQ,P+dP,a+da,α+dα,r++dr+) is

      f(M+dM,Q+dQ,P+dP,a+da,α+dα,r++dr+)=f(r)+fM|r=r+dM+fQ|r=r+dQ+fr|r=r+dr++fP|r=r+dP+fa|r=r+da+fα|r=r+dα,

      (50)

      where

      fM|r=r+=2r+,  fQ|r=r+=2Qr2+,  fr|r=r+=4πT,fP|r=r+=8πr2+3,  fα|r=r+=1r3ωq+1+,  fa|r=r+=1.

      (51)

      Using Eqs. (49), (50) and (51) yields

      dr+=2pr+r+da+r3ωq+dα4πr+(T2Pr+).

      (52)

      Then the variation of the entropy and volume is written as

      dS=2pr+r+da+r3ωq+dα2T4Pr+,

      (53)

      and

      dV=2r+pr+r2+da+r3ωq+1+dαT2Pr+.

      (54)

      From Eqs. (10), (53), (13) and (54), we obtain:

      TdSPdV=pr+r+2da+r3ωq+2dα.

      (55)

      Moreover, substituting Eqs. (12), (47) and (48) into Eq. (55), the relation between the internal energy and enthalpy is simplified to

      dM=TdS+VdP+ΦdQ+γdα+ϰda,

      (56)

      where γ and ϰ are physical quantities conjugated to the parameters α and a, respectively. They satisfy

      γ=12r3ωq+,ϰ=r+2.

      (57)

      Hence, the first law of thermodynamics is still satisfied.

      For an extremal black hole, the temperature is zero. Then Eq. (53) is modified as:

      dS=2pr+r+da+r3ωq+dα4Pr+.

      (58)

      If dα>0 and da>0, dS is less than zero and the entropy of the extremal black hole decreases over time. If dα<0 and da<0, dS could be greater than zero. Therefore, the second law of thermodynamics is indefinite for an extremal black hole in the extended phase space.

      For a near-extremal black hole, we analyse the change of entropy numerically to intuitively understand the changes in entropy. We set M=0.5 and l=pr=1. For the case ωq=2/3, a=0.01 and α=0.01, the extremal charge is Qe=0.465706962. When the charge is less than the extremal charge, the value of entropy changes when the value of charge changes. The values of r+ and dS corresponding to different values of charge are listed in Table 1.

      Q r+ dS da dα
      0.465706962 0.388699 −12.1128
      0.465706 0.389388 −12.2765
      0.46 0.440516 −61.7333
      0.44 0.495002 35.3525
      0.42 0.495002 20.7666 0.5 0.1
      0.40 0.550912 17.2452
      0.30 0.623442 12.1402
      0.20 0.661300 10.9798
      0.10 0.680975 10.5455

      Table 1.  The relation between dS, Q and r+.

      From Table 1, it can be seen that the event horizon of the black hole and the variation of entropy increases when the charge of the black hole decreases. It is obvious that there exists a phase transition point that divides the value of dS into positive and negative regions.

      In order to explore whether the values of state parameters of the cloud of strings and quintessence affect the second law of thermodynamics, we test the validity of the second law when the values of a, α and ωq change. In Table 2, we set a=0.1, with the values of the other variables the same as in Table 1. The extremal charge is Qe=0.480782137. The values of r+ and dS corresponding to different charge values are summarized in Table 2.

      Q r+ dS da dα
      0.480782137 0.407715 −11.5960
      0.480782 0.407979 −11.6471
      0.48 0.427511 −16.6251
      0.47 0.479200 −109.927
      0.45 0.524836 45.8092 0.5 0.1
      0.40 0.587213 19.7015
      0.30 0.653150 14.1111
      0.20 0.688780 12.6902
      0.10 0.707491 12.1425

      Table 2.  The relation between dS, Q and r+.

      In Table 3, we set ωq=1/2, a=0.1, α=0.1, with the values of other variables the same as in Table 1. The extremal charge is Qe=0.491516500. The values of r+ and dS corresponding to different charge values are summarized in Table 3.

      Q r+ dS da dα
      0.491516500 0.424248 −12.5738
      0.491516 0.424762 −12.6769
      0.49 0.452203 −20.8312
      0.47 0.525092 121.255
      0.45 0.560603 37.4749 0.5 0.1
      0.40 0.616308 20.7626
      0.30 0.678621 15.3953
      0.20 0.712972 13.8958
      0.10 0.731128 13.3019

      Table 3.  The relation between dS, Q and r+.

      In order to more intuitively observe the impact of a and α on dS, a function graph is used to express the relationship between dS and r+ in different situations, which is shown in Fig. 2. From Fig. 2, it is clear that there is indeed a phase change point that divides dS into positive and negative regions. If the charge of the black hole is less than the extreme value of the charge, the value of dS is negative and entropy decreases. If the charge is greater than the extreme charge, the value of dS is positive and the entropy increases. Thus, the second law of thermodynamics is indefinite for a black hole in the extended phase space. From Tables 1, 2 and 3, we find that when the value of the state parameter of the cloud of strings or quintessence increases, the extremal charge Qe and its corresponding r+ also increase. Moreover, for the same value of r+, the value of dS increases when values of a, α and ωq increase. The values of the parameters do affect the second law of thermodynamics, but the parameters do not determine whether the second law of thermodynamics is ultimately violated.

      Figure 2.  (color online) The relation between dS and r+, with parameter values M=0.5, l=pr=1, da=0.5, and dα=0.1.

      Furthermore, da and dα also affect dS. Changing da and dα, we analyze the change in dS numerically and plot it as shown in Fig. 3. Ultimately, it can be concluded that the second law of thermodynamics is not always valid for near-extremal black holes in the extended phase space.

      Figure 3.  (color online) The relation between dS and rh, with parameter values M=0.5, l=pr=1, ωq=2/3, a=0.1, and α=0.1.

    V.   OVERCHARGING PROBLEM IN EXTREMAL AND NEAR-EXTREMAL BLACK HOLES
    • In this section, the validity of the WCCC for extremal and near-extremal black holes is discussed by considering the absorptions of the scalar particle and fermion. An effective way to test the validity of the WCCC is to check whether the event horizon exists after the black hole absorbs the particles. The event horizon is determined by the function f(r). In the initial state, the minimum value of f(r) is negative or zero and f(r)=0 has real roots. This means the event horizon exists. When the black hole absorbs a particle, the mass and charge of the black hole change during the infinitesimal time interval dt and the minimum value of f(r) also changes. If the minimum value of f(r) is negative or equal to zero, the event horizon exists. Therefore, the WCCC is valid. Otherwise, the minimum value of f(r) is positive, and the event horizon does not exist. Consequently, the black hole is overcharged and the WCCC is not valid.

      The sign of the minimum value in the final state can be obtained in term of the initial state [100]. We assume (M,Q,P,r0,a,α) and (M+dM,Q+dQ,P+dP,r0+dr0,a+da,α+dα) represent the initial state and the final state, respectively. At r=r0+dr0, f(M+dM,Q+dQ,P+dP,r0+dr0,a+da,α+dα) is written as:

      f(M+dM,Q+dQ,P+dP,a+da,α+dα,dr0+r0)=δ+fM|r=r0dM+fQ|r=r0dQ+fP|r=r0dP+fa|r=r0da+fα|r=r0dα+fr|r=r0dr,

      (59)

      where

      fr|r=r0=0,  fM|r=r0=2r0,  fQ|r=r0=2Qr20,fP|r=r0=8πr203,  fa|r=r0=1,  fα|r=r0=1r3ωq+10.

      (60)

      At r=r0, we have

      f(M,Q,P,a,α,r0)f0=δ0,

      (61)

      and

      rf(M,Q,P,a,α,r0)fmin

      (62)

      From Eqs. (59), (60), (47), (48) and (54), we obtain:

      \begin{aligned}[b]f\left(M+{\rm d}M,Q+{\rm d}Q,P+{\rm d}P,a+{\rm d}a,\alpha+{\rm d}\alpha ,r_{0}+{\rm d}r_{0}\right) =& \delta-\frac{2Tp^{r}}{r_{0}\left(T-2Pr_{+}\right)}-\frac{2qQ}{r_{0}}\left(\frac{1}{r_{+}}-\frac{1}{r_{0}}\right)+\frac{8\pi}{3r_{0}}\left(r_{+}-r_{0}\right){\rm d}P\\& +\frac{Tr_{0}^{-3\omega_{q}}+2Pr_{+}\left(r_{+}^{-3\omega_{q}}-r_{0}^{-3\omega_{q}}\right)}{r_{0}\left(T-2Pr_{+}\right)}{\rm d}{\alpha}+\frac{Tr_{0}+2Pr_{+}(r_{+}-r_{0})}{r_{0}\left(T-2Pr_{+}\right)}{\rm d}a.\end{aligned}

      (63)

      When the initial black hole is an extremal black hole, r_{0} = r_{+} , T = 0 and \delta = 0 . Then we can obtain f_{\min} = \delta = 0 and f_{\min}^{\prime} = 0. Hence, Eq. (63) is written as:

      f\left(M\!+\!{\rm d}M,Q\!+\!{\rm d}Q,P\!+\!{\rm d}P,a\!+\!{\rm d}a,\alpha\!+\!{\rm d}\alpha ,{\rm d}r_{0}\!+\!r_{0}\right) \!=\! 0.

      (64)

      When the initial black hole is a near-extremal black hole, r_0 and r_+ do not coincide. In Eq. (63), the first term in the second line satisfies Eq. (61) and the second term is only suppressed by the test particle limit. However, the other terms are suppressed by the approaching extreme value limit and the test particle limit. Therefore, they can be ignored. Hence, Eq. (63) is modified as:

      \begin{aligned}[b]&f\left(M+{\rm d}M,Q+{\rm d}Q,P+{\rm d}P,a+{\rm d}a,\alpha+{\rm d}\alpha ,r_{0}+{\rm d}r_{0}\right) \\ =& \delta-\frac{2Tp^{r}}{r_{0}\left(T-2Pr_{+}\right)}<0. \end{aligned}

      (65)

      Hence, the near-extremal black hole stays near-extremal after absorbing a charged particle. The WCCC is satisfied for both extremal and near-extremal black holes in extended phase space.

    VI.   COMPARISON WITH OTHER WORKS
    • Recent studies of thermodynamics and the WCCC in extended phase space have attracted a lot of attention, but also aroused controversy. The controversy primarily focuses on the following point: when a particle with energy E is thrown into the black hole, the energy of the particle changes the internal energy of the black hole, i.e., E = {\rm d}U, or the energy of the particle changes the enthalpy of the black hole, i.e., E = {\rm d}M. In Ref. [57], the authors assume E = {\rm d}M and find the second law of thermodynamics is valid. In our work, we assume E = {\rm d}U and find the second law of thermodynamics is indefinite. When we assume E = {\rm d}M, we find the variation in the entropy of a RN-AdS black hole with cloud of strings and quintessence is:

      {\rm d}S = \frac{2p^{r}+2r_{+}^{3}l^{-3}{\rm d}l+r_{+}{\rm d}a+r_{+}^{-3\omega_{q}}{\rm d}\alpha }{2T}.

      (66)

      When {\rm d}\alpha > 0 and {\rm d}a > 0, the variation in the entropy is positive, though approaching zero in the extremal limit. When {\rm d}\alpha < 0 and {\rm d}a < 0, the variation in the entropy could decrease. Hence, the second law of thermodynamics is not always valid. Furthermore, we find when we assume E = {\rm d}M, f(M+{\rm d}M,Q+{\rm d}Q,P+{\rm d}P,r_0+{\rm d}r_0,a+{\rm d}a,\alpha+ {\rm d}\alpha) is rewritten as

      \begin{aligned}[b] &f\left(M+{\rm d}M,Q+{\rm d}Q,P+{\rm d}P,a+{\rm d}a,\alpha+{\rm d}\alpha ,r_{0}+{\rm d}r_{0}\right) \\ =& \delta-\frac{2p^{r}}{r_{0}}-\frac{2qQ}{r_{0}}\left(\frac{1}{r_{+}}-\frac{1}{r_{0}}\right)+\frac{8\pi r_{0}^{2}}{3}{\rm d}P-{\rm d}{\alpha}-\frac{1}{r_{0}^{3\omega_{q}+1}}{\rm d}a<0. \end{aligned}

      (67)

      Therefore, both extremal and near-extremal black holes cannot be overcharged after absorbing a charged particle. In future work, we plan to further study these two methods for various black holes.

    VII.   CONCLUSION
    • We have investigated the first and second laws of thermodynamics and the overcharging problem in a RN-AdS black hole with a cloud of strings and quintessence, via the absorption of scalar particles and fermions in extended phase space. The cosmological constant is treated as a function of thermodynamic pressure P. Moreover, the state parameters of cloud of strings and quintessence are treated as variables. To study the variations of the thermodynamic quantities of the black hole after absorbing a charged particle, we calculated the absorption of scalar particles and fermions. We found they finally simplified to the same relation p^{r} = \omega-q\Phi . This relation is exactly the same as that obtained by the Hamilton-Jacobi equation. The reason is that the Hamilton-Jacobi equation can be obtained by inserting the wave function (19) into the Klein-Gordon equation (18). The Hamilton-Jacobi equation takes the form:

      \left(\partial^{\mu}I+eA^{\mu}\right)\left(\partial_{\mu}I+eA_{\mu}\right)-m^{2} = 0.

      (68)

      In addition, the Hamilton-Jacobi equation can be obtained from the Dirac equation [101, 102]. The multiplication of Eq. (25) gives a second-order partial derivative equation. The wave function is written as \varPsi_{F} = \varPsi_{0}e^{\frac{i}{h}I} , where \varPsi_{0} is a position-dependent spinor. By inserting the function into the second-order partial derivative equation and keeping only the first-order term in \hbar , the Hamilton-Jacobian equation can be obtained.

      Usually, the thermodynamics and overcharging problem are discussed via absorbing a charged and/or rotating particle. However, a black hole can also absorb an uncharged particle and the thermodynamic properties of the black hole can be discussed in this way [103]. When the black hole absorbs an uncharged particle, only the mass of the black hole increases. Then there will be no case where the charge of the black hole is greater than the mass, the black hole cannot be overcharged and the WCCC is always satisfied. The following calculation also proves that this conclusion is correct. After the black hole absorbs an uncharged particle, q = 0 . The Klein-Gordon equation is written as

      \frac{1}{\sqrt{-g}}\frac{\partial}{\partial x^{\mu}}\left[\sqrt{-g}g^{\mu\nu}\frac{\partial}{\partial x^{\nu}}\right]\Psi_{S}-\frac{m^{2}}{\hbar^{2}}\Psi_{S} = 0.

      (69)

      The Dirac equation is modified as:

      i\gamma^{\mu}(\partial_{\mu}+\Omega_{\mu})\Psi_{F}+\frac{m_{0}}{\hbar}\Psi_{F} = 0.

      (70)

      Then Eqs. (24) and (42) become

      p^{r} = \omega.

      (71)

      Accordingly, the changes of the internal energy and charge of the black hole are written as

      {\rm d}U = p^{r},~~{\rm d}Q = 0.

      (72)

      At r_{+}+{\rm d}r_{+}, the final state of the black hole is represented by (M+{\rm d}M,P+{\rm d}P,r_{+}+{\rm d}r_{+}). Therefore, the first law of thermodynamics is:

      {\rm d}M = T{\rm d}S+V{\rm d}P+\mathcal{\gamma}{\rm d}\alpha+\varkappa {\rm d}a,

      (73)

      where

      \gamma = -\frac{1}{2r_{+}^{3\omega_{q}}},\varkappa = -\frac{r_{+}}{2}.

      (74)

      The first law of thermodynamics is the basic equation of the black hole system, and has nothing to do with whether or not particles are absorbed. We can derive this formula in this special way. The above equation is a special form of Eq. (56). The variation of the entropy is still

      {\rm d}S = \frac{2p^{r}+r_{+}{\rm d}a+r_{+}^{-3\omega_{q}}{\rm d}\alpha }{2T-4Pr_{+}}.

      (75)

      When the black hole is extremal, {\rm d}\alpha > 0 and {\rm d}a > 0, the value of dS is negative and the entropy of the black hole decreases. Hence, the second law of thermodynamics is violated.

      Furthermore, at r = r_0+{\rm d}r_0, f\left(M+{\rm d}M,P+{\rm d}P,a+{\rm d}a, \alpha+ {\rm d}\alpha ,r_{0}+ {\rm d}r_{0}\right) is written as

      \begin{aligned}[b] f\left(M+{\rm d}M,P+{\rm d}P,a+{\rm d}a,\alpha+{\rm d}\alpha ,r_{0}+{\rm d}r_{0}\right) =\ & \delta-\frac{2Tp^{r}}{r_{0}\left(T-2Pr_{+}\right)}-\frac{8\pi}{3r_{0}}\left(r_{+}^{3}-r_{0}^{3}\right){\rm d}P\\ &-\frac{Tr_{0}^{-3\omega_{q}}+2Pr_{+}\left(r_{+}^{-3\omega_{q}}-r_{0}^{-3\omega_{q}}\right)}{r_{0}\left(T-2Pr_{+}\right)}{\rm d}{\alpha}-\frac{Tr_{0}+2Pr_{+}(r_{+}-r_{0})}{r_{0}\left(T-2Pr_{+}\right)}{\rm d}a. \end{aligned}

      (76)

      For an extremal black hole, \delta = 0 , r_0 = r_+ and T = 0 . Then Eq. (56) is written as:

      f\left(M+{\rm d}M,P+{\rm d}P,a+{\rm d}a,\alpha+{\rm d}\alpha ,r_{0}+{\rm d}r_{0}\right) = 0.

      (77)

      Therefore, extremal black holes cannot be overcharged.

      For a near-extremal black hole, we define r_{+} = r_0+\epsilon , where 0<\epsilon\ll1 . Then Eq. (56) is written as:

      \begin{aligned}[b] f\left(M+{\rm d}M,P+{\rm d}P,a+{\rm d}a,\alpha+{\rm d}\alpha ,r_{0}+{\rm d}r_{0}\right) =\ & \delta-\frac{2Tp^{r}}{r_{0}\left(T-2P\left(r_{0}+\epsilon\right)\right)}-8\pi r_{0}\epsilon {\rm d}P\\ &-\frac{Tr_{0}^{-3\omega_{q}}+2Pr_{0}\epsilon\left(-3\omega_{q}\right)r_{0}^{-3\omega_{q}-1}}{r_{0}\left(T-2P\left(r_{0}+\epsilon\right)\right)}{\rm d}{\alpha}-\frac{Tr_{0}+2Pr_{0}\epsilon}{r_{0}\left(T-2P\left(r_{0}+\epsilon\right)\right)}{\rm d}a+O\left(\epsilon^{2}\right). \end{aligned}

      (78)

      When the initial black hole is near-extremal, we have {\rm d}P\sim\epsilon, {\rm d}a\sim\epsilon and {\rm d}\alpha\sim\epsilon. Hence, the above equation leads to

      \begin{aligned}[b] &f\left(M+{\rm d}M,P+{\rm d}P,a+{\rm d}a,\alpha+{\rm d}\alpha ,r_{0}+{\rm d}r_{0}\right) \\ =& \delta-\frac{2Tp^{r}}{r_{0}\left(T-2Pr_{+}\right)}<0. \end{aligned}

      (79)

      Therefore, near-extremal black holes cannot be overcharged after absorbing an uncharged particle. Moreover, an extremal black hole stays extremal and a near-extremal black hole stays near-extremal.

      We used the relation p^{r} = \omega-q\Phi to recovered the first law of thermodynamics and discussed the validity of the second law of thermodynamics and the WCCC. During the discussion, the final state of the black hole was considered to be still a black hole. The first law of thermodynamics is recovered and the second law of thermodynamics is indefinite. Since we treated the parameters related to the cloud of strings and quintessence as variables, there are two more terms, -\dfrac{1}{2r_{+}^{3\omega_{q}}} {\rm d}\alpha and -\dfrac{r_{+}}{2} {\rm d}a, in the formula of the first law of thermodynamics. The detailed formulas of the first law of thermodynamics for black holes and black holes surrounded by clouds and quintessence are shown in Table 4.

      Type of black hole1st law in extended phase space
      RN-AdS BH{\rm d}M=T{\rm d}S+V{\rm d}P+\varphi {\rm d}Q
      RN-AdS BH with cloud of strings and quintessence{\rm d}M=T{\rm d}S+V{\rm d}P+\varphi {\rm d}Q-\dfrac{1}{2r_{+}^{3}\omega_{q} }{\rm d}\alpha-\dfrac{r_{+} }{2}{\rm d}a

      Table 4.  Results for the first thermodynamic law under different conditions in extended phase space.

      When testing the validity of the second law of thermodynamics, we found there exists a phase change point that divides the value of dS into positive and negative value regions. The change in the values of the state parameters related to the cloud of strings and quintessence will affect the value of r_+ , Q_e and dS. As shown in Fig. 2, the parameters do affect the second law of thermodynamics, but the parameters do not determine whether the second law of thermodynamics is ultimately violated. Moreover, the values of da and {\rm d}\alpha also affect the values of Q_e and dS. Furthermore, the WCCC has been proven to be valid all the time for extremal and near-extremal black holes. The validity of the WCCC was tested by checking the sign of the minimum value of f(r) . Compared with black holes without a cloud of strings and/or quintessence, the minimum value of f(r) becomes larger after absorbing particles with energy and charge. However, the minimum value of f(r) remains the original positive and negative. Therefore, neither extremal black holes or near-extremal black holes will be overcharged. Our results are shown in Table 5.

      1st lawSatisfied
      2nd lawIndefinite
      WCCCSatisfied for extremal and near-extremal black holes. Extremal/near-extremal black holes stay extremal/ near-extremal after charged test particle absorption

      Table 5.  Results for the first and second laws of thermodynamics and the WCCC, tested for a RN-AdS black hole with cloud of strings and quintessence via charged test particle absorption.

      In this paper, to satisfy the Smarr relation, the parameters associated with cloud of strings and quintessence are consider as variables. In previous studies, another approach was also taken. The parameters related to the cloud of strings and quintessence were considered as constants and the thermodynamic properties of black holes investigated [58]. When the terms related to a and \alpha are turned off, in the extended phase space, the cosmological constant is treated as a variable and the parameters related to the cloud of strings and quintessence are treated as constants. The first law of thermodynamics is:

      {\rm d}M = T{\rm d}S+\varphi {\rm d}Q+V{\rm d}P.

      (80)

      The variation of the entropy is:

      {\rm d}S = \frac{2p^{r}}{2T-4Pr_{+}}.

      (81)

      The denominator of dS has a negative value for an extremal black hole, which means the entropy of the black hole decreases at least in the extreme case. Hence, the second law of thermodynamics is violated.

      Furthermore, at r_0+{\rm d}r_0, f(M+{\rm d}M,Q+{\rm d}Q,P+{\rm d}P, r_{0}+ {\rm d}r_{0}) is:

      \begin{aligned}[b]& f\left(M+{\rm d}M,Q+{\rm d}Q,P+{\rm d}P,r_{0}+{\rm d}r_{0}\right) \\ =& \delta-\frac{2Tp^{r}}{r_{0}\left(T-2Pr_{+}\right)}-\frac{2qQ}{r_{0}}\left(\frac{1}{r_{+}}-\frac{1}{r_{0}}\right)-\frac{8\pi}{3r_{0}}\left(r_{+}^{3}-r_{0}^{3}\right){\rm d}P < 0.\end{aligned}

      (82)

      Therefore, both extremal and near-extremal black holes cannot be overcharged.

    ACKNOWLEDGMENTS
    • We are grateful to Peng Wang, Haitang Yang, Deyou Chen and Xiaobo Guo for useful discussions.

Reference (103)

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