Thermodynamic phase transition for Quintessence Dyonic Anti de Sitter Black Holes
Hossein Ghaffarnejad Emad Yaraie Mohammad Farsam









10mm equationsection

    Thermodynamic phase transition for Quintessence Dyonic Anti de Sitter  Black Holes
     0.5 cm
       Hossein Ghaffarnejad [E-mail: hghafarnejad@semnan.ac.ir
     ] Emad Yaraie [E-mail: eyaraie@semnan.ac.ir
     ] and Mohammad Farsam [E-mail: mhdfarsam@semnan.ac.ir
     ] 0.1 cm Faculty of Physics, Semnan
    University, P.C. 35131-19111, Semnan, Iran 
    
    We study thermodynamic of a dyonic AdS black hole surrounded by
    quintessence dark energy where negative cosmological constant of
    AdS space behaves as pressure of the black hole. We choose grand
    canonical ensemble of the black hole where its   magnetic charge
    Q_M  and electric potential Φ_E are hold as constant. Our
    goal  in this work is study physical effects of the magnetic
    charge and electric potential on the thermodynamic phase
    transition of the black hole in presence of quintessence dark
    energy. When barotropic index of the quintessence is
    ω=-7/9 we obtained that compressibility factor of
    the black hole reduces to Z_c=3/8 which corresponds to
    the Van der Waals fluid. We obtained analogy between the
    small/large black hole phase transition and liquid/gas phase
    transition of the Van der Waals fluid. Numerical calculations
    predict that the black hole may born plasma phase which is
    fourth different state of the matter which does not appear in the
    Van der Waals fluid.
    
    
    § INTRODUCTION
      Thermodynamic aspects of black holes have  challenge when the
    argument of thermodynamic phase transition is under consideration.
    This is because of association of the phase transition to the
     thermodynamic variables for instance entropy, volume  and etc. One
    of most controversial thermodynamic variables is volume of black
    holes undoubtedly. Consider a flat space-time including a black
    hole and an observer  located at distances far from the black hole
    which can not observe small region of space-time hidden behind the
    black hole horizon. In fact this region which is equal with
    geometric volume of the black hole doesn't have an straightforward
    definition due to use of different gravity models. It is well
    known that nonzero value of the vacuum energy density of the de
    Sitter space is relating to the cosmological constant. In the
    latter case a typical definition for the volume is there: That is
    related to total energy of the gravitational system but not to the
    black hole mass amount which is stand as hidden from point of view
    of an observer far from the black hole (ADM mass [1,2,3]). This
    means that the hidden energy from point of view of observer would
    be part of total energy which is equal to the ADM mass located on
    hidden volume. In the thermodynamic of the black holes the
    observed mass by an observer far from the black hole is considered
    to be the black hole thermal energy while the cosmological
    constant behaves as a fixed parameter. For an anti de-Sitter space
    the cosmological constant has negative values arising from an
    energy-momentum tensor which reads the vacuum equation of state
    P=-ρ=-Λ/8π G_N. Regarding the latter concept
    one can write first law of black holes thermodynamic as follows.
    
        dM=TdS+VdP  ⇒  dU=TdS-PdV
    
    where U=M-PV and H=U+PV are internal energy and enthalpy of
    the black holes respectively. This  shows the black hole mass M
    can interpreted as enthalpy. In this view the cosmological
    constant is related to pressure of the AdS space-time [4]. Since
    the cosmological constant corresponds to space-time pressure thus
    its conjugate variable must have volume dimension. This is called
    thermodynamic volume for which we have V=(∂
    M/∂ P)_S [5]. In general, this definition of the black
    hole volume is different with the geometric volume which we
    pointed previously. This contains some universal properties which
    they satisfy the condition of reverse iso-perimetric inequality
    [6]. Regarding the above definition for the black hole
    thermodynamic volume and the pressure, one can study critical
    behavior of the black holes in the P-V frame and in the extended
    phase space [7] (see also [8] for charged and [9] for rotating
    black holes). For a dyonic black hole, authors of ref. [10]
    studied magnetic charge effects on T-V critically of the black
    holes via the holographic approach. They observed that in constant
    electric potential and constant magnetic charge, the phase diagram
    of ensemble of dyonic black hole is similar to well known phase
    transition of Van der Waals fluid done in presence of its internal
    chemical potential. They obtained ferromagnetic like behavior of
    boundary theory of the dyonic black hole when the external
    magnetic field vanishes. Authors of the work [11] studied phase
    structure of the quintessence Reissner-Nordström-AdS black
    hole with the nonlocal observables such as holographic
    entanglement entropy and two point correlation functions. Results
    of their work show that, as the case of the thermal entropy, both
    the observables exhibit the similar Van der Waals-like phase
    transition. They were check the equal area law  and critical
    exponent of the heat capacity for the first and the second order
    phase transition respectively. Also they discussed the effect of
    the parameter of state equation on the phase structure of the
    nonlocal observables. Hartnoll et al studied AdS/CFT
    superconductor properties of a planar AdS Schwarzschild black hole
    which for temperatures less than the critical temperature, reduces
    to  a charged condensate state. This is formed via a second order
    phase transition for which the conductivity reaches to infinite
    value [12,13, 14,15,16]. The study of the  phase transition of the
    dyonic black hole surrounded by dark energy must be very complex
    and attractive because of dependence to the possibility existence
    of the dark energy. In fact There is a wide range of modern
    cosmological observations which confirm that our universe expands
    with positive acceleration. This leads us to infer that instead of
    the Newton`s gravity force there should exist other forces which
    is not still observed [17,18,19]. Dark energy is one of the most
    important candidates to explain this situation, which must
    occupied seventy percent of energy of our universe. Among diverse
    dark energy models, `quintessence` as a canonical scalar field can
    be a good model to explain acceleration of the universe
    [20,21,22]. There are many works where various gravitational
    models are used to study thermodynamic aspect of black holes  (see
    for instance [23-26]).
    
    
    In this work we study thermodynamics of the AdS dyonic black hole
    which is surrounded by the quintessence dark energy.
     We calculate
    its equation of state on the event horizon hypersurface where the
    cosmological constant behaves as the black hole pressure. Then for
    different values of barotropic index of the quintessence dark
    energy ω we obtained parametric forms of critical points
    {T_c,V_c,P_c} defined in terms of the constant electric
    potential and magnetic charge and ω. Then we set
    ω=-7/9 to obtain numerical values of the critical
    points. Because for this particular value of the quintessence
    barotropic index the black hole compressibility factor reaches to
    one which is obtained for the Van der Waals fluid as
    Z=3/8. This restriction helps us to improve statement
    which  extract from the work.
     The organization of the work is as follows.
     In section 2 we review
    briefly phase transition of a Van der Waals fluid.
     In section 3 we define a dyonic AdS black hole in presence of quintessence dark
    energy counterpart. Then we obtain its equation of state by
    solving equation of the event horizon. Enthalpy, heat capacity at
    constant pressure and Gibbs free energy are other thermodynamic
    variables of the black hole which we calculated in this section.
    At last we plot variations of the these variables versus the
    volume and the temperature. Also we obtained all possible
    numerical values for the critical points at zero electric
    potential Φ_E=0 versus different values of the magnetic
    charge q_M. Section 4 denotes to results of the this work and
    outlooks.
    
    
    § PHASE TRANSITION OF VAN DER WAALS FLUID
       Equation of state for an ideal gas has simple form as
    PV=Nk_BT in which for a constant temperature (isotherm curves)
    the pressure decreases absolutely by increasing the volume. While
    this is not ready for a real `Van der Waals` fluid. In the latter
    case, the equation of state  given by the following form behaves
    complex [27].
    
        P(T,V)=Nk_BT/V-Nb-aN^2/V^2,
    
     where N is number of particles in the fluid and the constant b is intended to correct for the volume occupied by the molecules and the
      term aN^2/V^2 is a correction that accounts for the intermolecular forces
     of attraction. In fact, these constants are evaluated by noting that the critical isotherm passes through a
      point of inflection at the critical point and that the slope is zero at this point.
     The critical point {T_c,V_c,P_c} is determined by solving
     
        (∂ P/∂ V)_T=(∂^2
        P/∂ V^2)_T=0
    
    which read
    
        V_c=3bN,   P_c=a/27b^2,   k_BT_c=8a/27b
     and one can calculate compressibility dimensionless factor Z_c
    as follows.
    
        Z_c=P_cV_c/Nk_BT_c=3/8.
    
    Substituting
     the dimensionless thermodynamic variables p=P/P_c, v=V/V_c, t=T/T_c one can infer the equation of state
     (2.1) reads to a dimensionless form as follows (see ref. [27]).
    
        p(v,t)=8t/3v-1-3/v^2.
    
    Gibbs free energy difference for a Van der Waals fluid is given by
    
        dG=-SdT+VdP
     where S
     is entropy. For a constant temperature the first term in right hand side of the above equation is eliminated and so we
    obtain 
        G=∫ VdP=PV-∫ P(V)dV
    
    which for the dimensionless equation of state (2.5) reads
    
        g(t,v)=8t/3[3v/3v-1-ln(3v-1)]-6/v.
    
    Holding the temperature as constant, we plot diagrams for the
    equation of state (2.5) and for the Gibbs free energy (2.8) in
    figures 1-c and 1-b respectively. Here t=0.5<t_c,t=t_c=1 and
    t=1.5>t_c correspond to dash, solid and dash-dot lines
    respectively.  (1-c) shows for t<t_c there is a phase transition
    for the fluid which coexistence of two phase occurs at the minimum
    pressure. This point has minimum Gibbs free energy which is shown
    in figure (1-b). Figure (1-a) shows variation of the Gibbs free
    energy versus the temperature. It shows at all, the Van der Waals
    fluid has different phases which they cross each other at the
    phase transition point. This point corresponds to coexistence
    regime of both the gas and the fluid phases of the Van der Waals
    matter which occurs at small scale of thermodynamic volume.
    
    We obtained by numerical calculations that for p≥ p_c=1 the
    diagrams of p-v, g-t and or g-v have not a minimum point in
    phases space (not shown) and so the phase transition dose not
    appeared in the latter case
    .
    
    Our aim in this work is to study thermodynamic phase transition of
    a quintessence AdS dyonic black hole and obtain situations for
    which the black hole behaves thermodynamically as a Van der Waals
    fluid.
    
    
    §  QUINTESSENCE DYONIC ADS BLACK HOLE THERMODYNAMIC
     According to the work presented by Kiselev in [23] for metric of a
    spherically symmetric static Reissner Nordström black hole
    surrounded by quintessence dark energy we use an extension of his
    metric solution where an extra magnetic charge Q_M is added and
    so it is called as dyonic AdS quintessence black hole with the
    following metric form.
    
        ds^2=f(r)dt^2-dr^2/f(r)-r^2(dθ^2+sin^2θ
        dϕ^2),
    
    in which for quintessence regimes -1<ω<-1/3 we have
    
        f(r)=1-2M/r+Q_M^2/r^2+Q_E^2/r^2-(r_q/r)^1+3ω-Λ r^2/3,
    
    where r_q  is the dimensional normalization constant and the
    negative  cosmological constant
     Λ<0 of the AdS space behaves as thermodynamic pressure of the black hole. Quintessence counterpart of the above metric solution
     (r_q/r)^1+3ω comes from leading order terms of a convergent
     series solution  which is obtained originally by the Kiselev by applying the condition of linearity and additivity of the Einstein
     metric equation [23]. 
    
    As we know the phase structure depends on the statistical ensemble
    which we choose to study the phase transition. There is no new
    interesting results if the black hole phase transition is studied
    by particular ensemble in which both of the electric and the
    magnetic charges are hold as constant.  In the contrary if we hold
    magnetic charge  together with the electric potential as constant
    parameters then there is obtained possibly new physics from study
    of the phase transition.
     Former case is a canonical ensemble where a
    small/large black hole phase transition dose not appeared in
    absence of the quintessence while the latter setting become grand
    (mixed) canonical ensemble for which a small/large black hole
    phase transition occurs [10, 28]. However the former situation is
    similar to case where the electric charge is shifted by Q_E^2→
    Q_E^2+Q_M^2 while in the latter case the first law of
    thermodynamics is described in extended phase space (see for
    instance [29]).
     To approach our goal in this work we extend the first
     law of the black hole thermodynamic given by the equation (2.12)
     in ref. [30] as follows.
    
    
    
    
        dM=TdS+Φ_EdQ_E+Φ_MdQ_M+VdP+Ad r_q
    
    where A is conjugate variable for the normalization factor r_q
    of the quintessence dark energy. It is defined by
    
        A=(∂ M/∂ r_q)_S,Q_E,Q_M,P.
    
     The Smarr relation is obtained from the mass function of the black hole because the black hole mass is equal to
      the enthalpy. Applying the black hole metric (3.2) and
      solving the horizon equation f(r_h)=0 we obtain mass of the AdS quintessence dyonic black hole versus the thermodynamics variables
      as follows.
     
        M=r_h/2+Q_M^2/2r_h+Q_E^2/2r_h-r_q^1+3ω/2r_h^3ω+4π r_h^3/3P
    
    where we substitute pressure P of the AdS space with
    corresponding radius r_AdS as follows.
    
        P=-Λ/8π,    r_AdS=√(-3/Λ).
    
    Using (3.3) one can calculate thermodynamic volume of the
    quintessence dyonic AdS black hole as
    
        V=(∂ M/∂ P)_A,Q_E,Q_M,r_q=4π r_h^3/3
    
    which its form is similar to the geometric volume of the horizon
    with surface era 4π r_h^2. We should notice that the latter
    statement is not correct for metric of an arbitrary curved space
    time and in general the black hole thermodynamic volume is differ
    with its geometric one. From (3.5) we obtain
    
        Φ_E=(∂ M/∂ Q_E)_S,P,Q_M,r_q=Q_E/r_h
    
     which is called as the electric potential of the black hole on the horizon.
     Substituting (3.5) and calculating (3.4) we obtain
     
        A=-(1+3ω)/2(r_q/r_h)^2ω.
    
     The Hawking temperature of the above black hole can be derived as follows.
     
        T=f^'(r)/4π_|_r=r_h=1/4π(1/r_h-Q_M^2/r_h^3-Q_E^2/r_h^3+3ω
         r_q^1+3ω/r^3ω+2_h+8π r_h P
         )
     where we substitute (3.5) to remove M. Since the black hole mass M is interpreted as enthalpy as H=M
     thus one can infer that the equation (3.5)
      reads to the following form.
      
        H=U+PV
     where
       
        U=r_h/2+Q_M^2/2r_h+Q_E^2/2r_h-r_q^1+3ω/2r_h^3ω
    
       is internal energy. To study location of critical point on the P-V plan it is appropriate to use dimensionless forms of the thermodynamic
       functions as follows.
          
        p=8π r_q^2P,   t=4π r_qT,   v=r_h/r_q,    q_M=Q_M/r_q
     Substituting
     the above definitions and (3.8) into the equation (3.10) we obtain dimensionless equation of state for
     the quintessence dyonic AdS black hole such that
    
        p(v,t)=t/v-(1-Φ_E^2)/v^2+q_M^2/v^4-
        3ω/v^3(1+ω).
    
    By applying the above dimensionless parameters the  enthalpy
    (3.11) reads
    
        m=(1+Φ_E^2)v+q_M^2/v-1/v^3ω+pv^3/3
    
     in which
    
        m=M/2r_q
    
    is assumed to be dimensionless mass of the black hole. Solving the
    equations
    
        (∂ p/∂
        v)_other parameters=(∂^2 p/∂
        v^2)_other parameters=0
     we can obtain parametric form of the
    critical point {p_c,t_c,v_c} which satisfy the following
    equations.
    
        2(1-Φ_E^2)v_c^3ω+1-12q_M^2v_c^3ω-1+9ω(1+ω)(2+3ω)=0
    
        t_c=2(1-Φ_E^2)/v_c-4q_M^2/v_c^3+9ω(1+ω)/v_c^2+3ω
    
        p_c=(1-Φ_E^2)/v_c^2-3q_M^2/v_c^4+3ω(2+3ω)/v_c^3(1+ω)
        .
       The above algebraic equations have several roots
    due to different values of the black hole parameters
    {ω,q_M,Φ_E} which are into account where
    -1<ω<-1/3 corresponds to the quintessence dark
    energy regime.  To compare compressibility factor of this black
    hole with which one obtained for the Van de Waals fluid as
    Z=3/8 given by the equation (2.4), it is appropriate to
    substitute 1-Φ^2_E from (3.18) into the equations (3.19) and
    (3.20). In the latter case (3.19) and (3.20) reduce to the
    following forms respectively. 
        t_c=8q_M^2/v_c^3-9ω(1+ω)^2/v_c^2+3ω
    
    and
    
        Z=p_cv_c/t_c=3/8-3ω(1+3ω)(7+9ω)/8t_cv_c^2+3ω.
    
    The above relation shows that for a quintessence regime of the
    dark energy where -1<ω<-1/3 the compressibility of
    the dyonic AdS black hole behaves as a Van der Waals fluid if we
    set
    
        ω_Van=-7/9.
      Substituting (3.23) into the equations
    (3.14), (3.18), (3.21) and (3.22) we will have respectively
    
        p=t/v-(1-Φ^2_E)/v^2+q_M^2/v^4+7/31/v^2/3
    
        1-Φ_E^2=6q_M^2/v_c^2-7/27v_c^4/3,
    
        t_c=8q_M^2/v_c^3+28/81v_c^1/3
    
    and
    
        p_c=3q_M^2/v_c^4+7/541/v_c^2/3.
    
    The equations (3.26) and (3.27) show that the critical temperature
    t_c and the critical pressure p_c are sensitive to the
    magnetic charge of the black hole q_M just at small values of
    the critical volume v_c<<1 but not for
    its large amounts. 
    
    For positive temperatures t≥0 (here  we exclude negative
    temperatures because they are in usual way un-physical) one can
    infer that p-v diagram plotted by the equation (3.24) possess
    inflection points under condition
    
        0≤Φ_E^2<1
     and these inflection points disappear for
    electric potentials Φ_E^2≥1. Because for (3.28)
    coefficient of the term 1/v_c^2 given by (3.24) remains
    negative. Substituting
     (3.28) into the equation (3.25) we can obtain
    admissible numerical values for the critical volume as
    
        0<6q_M^2/v_c^2-7/27v_c^4/3≤1.
    
    This shows upper and lower bounds of the critical volume v_c
    which are become restricted by the magnetic charge q_M. For
    simplicity we choose equal sign in the above inequality identity
    for which we will have
    
        q_M(Φ_E=0)=±v_c/18√(14v_c^4/3+54).
    
    Substituting (3.30) into the relations (3.26) and (3.27) we obtain
    
        t_c=81+77v_c^4/3/162v_c
     and
    
        p_c=14v_c^2+27v_c^2/3/54v_c^8/3
    
    for zero electric potential Φ_E=0. Substituting (3.30) and
    Φ=0 into the equation (3.24) we obtain
    
        p=t/v+v_c^2/v^4(1+7/27v_c^4/3)
        [1/6-(v/v_c)^2]
        +7/31/v^2/3[9+(v_c/v)^4/3].
    
    This shows that the for some positive constant temperatures p-v
    diagrams will have inflection points only for
    
        v/v_c>1/√(6)
    
    because of negativity sign of the term
    [1/6-(v/v_c)^2]. Now we talk
    about the enthalpy function (3.15) which under the condition
    (3.30) reads
    
        m=(2+7/27v_c^4/3)v+v_c^2/v(1+7/27v_c^4/3)[1/6-
        (v/v_c)^2]-1/v^7/3+pv^3/3.
    
     Other
    quantity which can be used to study thermal stability of black
    holes is heat capacity at constant pressure defined by
    
        C_P=(∂ M/∂ T)_P
     which for C_p<0 the black hole is
    unstable while for C_p>0 is stable. Applying (3.33) and (3.35)
    the equation (3.36) reads
    
        c_p=(∂ m/∂ t)_p=W(v)/O(v)
    
    in which W(v) and O(v) are obtained respectively as follows.
    
        W(v)=(∂ m/∂
        v)_p=2+pv^2+7/31/v^10/3-v_c^2/v^2(1+
        7/27v_c^4/3)[1/6+(v/v_c
        )^2]
     and
    
        O(v)=(∂ t/∂
        v)_p=p+v_c^2/v^4(1+7/27v_c^4/3)[1/2-(v/v_c)^2]+7/31/v^2/3[(v_c/v)^4/3-3]
    
    with
    
        c_p=C_P/8π r_q^2.
    
    Gibbs free energy of the quintessence dyonic AdS Black hole is
    given  by (see Eq. (3.2) in ref. [10])
    
        G=M-TS-Φ_EQ_E-Φ_MQ_M=μ N
    
    where S is entropy of the black hole which by according to the
    Bekenstein-Hawking entropy formula is quarter of the black hole
    horizon area [31] (see also [32]).
      
        S=∫dM/T=∫_0^r_hdr_h/T(∂ M/∂ r_h)
          =4π r_h^2/4=π r_h^2=horizon  area /4.
    M is the black hole mass which is equal to the black hole
    enthalpy energy. T is the black hole Hawking temperature and
    Φ_E,M and Q_E,M are electromagnetic potential and
    electromagnetic charge of the black hole. N is number of micro
    particles and μ is their mutual chemical potential.
    Substituting (3.5), (3.8), (3.10) and (3.39) into the Gibbs free
    energy (3.38) one can infer
      
        G=(1-Φ^2_E)r_h/4-1/4Q_M^2/r_h-2π r_h^3 P/3-(2+3ω)r_q^1+3ω/4
          r_h^3ω
    
     which by substituting (3.13)  leads to a dimensionless form as follows.
    
        g(p,v)=(1-Φ_E^2)v-q^2_M/v-(2+3ω)/v^3ω-pv^3/3
     where we  defined dimensionless Gibbs free energy
    as follows.
    
        g=4G/r_q.
    
    Substituting (3.23), (3.25) and (3.30) with Φ_E=0 into the
    black hole Gibbs free energy (3.44) we obtain
    
    
        g(p,v)=v_c^2/v(1+7/27v_c^4/3)[(v/v_c)^2-
        1/6]+7/27v[9-(v_c/v)^4/3]-pv^3/3
    
    Now we plot diagrams of state equation of the black hole p(v,t),
    its Gibbs free energy g(p,v) versus the volume and the
    temperature for different values of the pressure and then proceed
    to interpretation of these diagrams. Diagrams of the thermodynamic
    variables such as the pressure (3.33), the enthalpy (3.35), the
    heat capacity (3.37), and Gibbs free energy (3.46) together with
    all possible numerical values of the critical points of the
    quintessence AdS dyonic black hole are plotted in figures 2 and
    3.
    
    Diagram of figure (2-a) shows all possible values for critical
    volume obtained from the equation (3.25) by substituting |q_M|>0
    and 0≤Φ_E^2<1. Setting Φ_E=0 we plot all possible
    values for (q_M,v_c) by applying (3.30) in figure (2-b). The
    figure (2-c) denotes variation of the critical compressibility of
    the black hole z_c=p_cv_c/t_c which can be calculated
    from (3.31) and (3.32) versus the critical volume v_c. This
    figure shows that for large v_c the black hole system takes a
    single phase, while for small v_c the black hole system is made
    from to subsystem (phase) with different compressibility factor
    which they reach to a coexistence stable state with smallest value
    of v_c. Figure (2-d) describes variations of the critical
    temperature versus the critical volume containing a local minimum
    value. In other words by decreasing and increasing the critical
    volume the critical temperature raises. We show absolutely
    decreasing behavior of the critical pressure by raising the
    critical volume in the figure (2-e). Applying (3.31) and  (3.32)
    we plot possible values of the  critical pressure against the
    possible values of the  critical temperature in the figure (2-f).
    This shows monotonically raising behavior for p_c when the
    temperature increases t_c for approximate values
    t_c≽2300 but not for lesser values. For small values of
    the critical temperature, p_c increases rapidly. Diagram of the
    equation (3.33) is plotted in figure (2-g) at constant pressure.
    This figure shows a local maximum and a local minimum critical
    temperature for small and large volume of the black holes
    respectively. Looking to this figure one can infer that for
    temperatures 3.5<t<5.8 there is three unstable state for the
    black holes under consideration. Minimum of the temperature
    diagram  appears at v≈0.18 and all black holes with
    v<0.18 is unstable and so they are hotter. They incline to reach
    a cooler larger stable state thermally with volume ∼0.18.
    This means small/large black hole phase transition. Also all black
    holes with v>0.18 incline to compress so that become smaller one
    with stable volume ∼0.18. This is also means that unstable
    large black holes exhibit with a large/small phase transition.
    This phase transition can be interpreted in figure (2-h) at
    constant temperature for a given pressure -20<p<60. This domain
    is depended to some fixed temperatures which we used to plot the
    diagram. One can extract more physics from behavior of the black
    hole by looking the figure (2-i). In contrary the figure (1-a) for
    the Van der Waals fluid which has a single crossing point the
    figure (2-i) predicts a double crossing points. A single crossing
    point at (1-a) shows a coexistence  of a fluid containing two
    subsystems or phase (gas/fluid),while (2-i) predicts that the
    black hole matter contains two coexistence state because of
    existence two crossing point. This shows that the black hole can
    have 4 subsystems or phase. Importance of the diagram (2-i) with
    respect to other diagrams (2-g) and (2-h) is this: We saw that by
    setting t=constant and p=constnat in the figures (2-g) and
    (2-h) there is obtained three particular points with different
    volumes which predict three unstable state of the black hole
    matter. Physically these 3 unstable state can be considered as
    gas/fluid/solid of phase the black hole material. They do not
    predict forth plasma phase of the matter which can be coexistence
    with other phases of the black hole matter.  In contrary the
    figure (2-i) namely variation of the black hole Gibbs free energy
    versus the temperature can be predict the plasma phase of the
    black hole matter which can be occurs under the particular
    conditions.
    
    
    Diagrams of the figures (3-a) and (3-b) show the above situations
    via variations of the heat capacity of the black hole at constant
    pressure versus the volume and the temperature respectively. We
    know that a thermodynamical system become unstable when its heat
    capacity takes negative values and stable state for its positive
    values. These diagrams show that this black hole will be stable
    C_p>0 thermally just when its volume is not more larger or
    smaller than the critical volume v_c=0.1. There is small volume
    t_s<0.1 and large volume v_L>0.1 where sign of the heat
    capacity is changed. This is also predicts only a small/large
    black hole phase transition which can be contain three phase of
    the black hole matter. Diagram of the figure (3-b) shows a phase
    of the black hole matter which by raising the temperature the heat
    capacity has zero value C_v≈0 while two other phases of
    the black hole matter behave as heat-giveier(C_p<0) and
    heat-receiver(C_p>0). There is a particular temperature
    t→1.5 where the heat capacity takes a degenerate state. Last
    diagram (3-c) shows the enthalpy which has positive and negative
    values for large and small black holes respectively. When we say
    black hole is small
    then is should be comparable with the critical volume v_c=0.1.
    
    Last point which should be presented is this: All diagrams are
    plotted easily except (g-t). This has more sensitive to initial
    values. For instance with p=30 and 0<v<1 the diagram shows the
    black hole have two crossing points  but for  the number of these
    crossing points can be reduced by changing the pressure.  Our
    numerical calculations predicts they can be even disappeared by
    changing the volume domain and the pressure. However as future
    work, it's worth looking at finding more points of this type which
    predict different phases of the black hole material. If there is
    obtained more than 2 crossing point then one can result that the
    black hole  exhibits with other  phases of the matter called as
    the `Bose-Einstein  condensate` state and the `Fermionic
    condensate` state which are fifth and sixth superfluid phase of
    the matter respectively [34,35].
    
    
    
    
    
    
    
    § CONCLUSION
     In this work we studied quintessence dark
    energy effects on the phase transition of the AdS dyonic black
    holes. We determined regimes on the barotropic index of the
    quintessence dark energy and electric potential and magnetic
    charge of the black hole where the small/large black hole phase
    transition is occurred. Also diagram of the p-v curves predicts
    for large values of the thermodynamic volume of the black hole
    phase transition can be reach to the well known Hawking-Page phase
    transition where the black hole disappears and a vacuum AdS space
    remains. Our work other phases of the black hole matter is
    predicted such as plasma,Bose-Einstein and Fermionic
    condensate in addition to three well known states called as
    solid-fluid-gas of the matter.  One of applicable work which we
    can be investigate it in the future is study quintessence dark
    energy effects on entanglement entropy of AdS dyonic black holes
    in relation to the Maxwell equal area law which does not satisfied
    for  AdS dyonic black holes  [28].
       1cm
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