A consistent thermodynamic treatment for quark mass density

							A Consistent Thermodynamic
 Treatment for Quark Mass
 Density-Dependent Model
         Ru-Keng Su
      Physics Department
       Fudan University
              Difficulties
 In relativistic energy dispersion relation:

   ( p)  m  p  m (T ,  )  p
               2     2         *        2       2


 Ω becomes an explicit function of m:

        Ω  Ω(T ,V , {i }, m (T ,  ))
                               *


 How will the thermodynamic formulae with
  the partial derivatives become?
Different treatments with extra terms
         from partial derivatives

A.
                           B 
                      p                             ,
      B.
                         V V  B         T ,{  i }

                B                             T 
                                    i  i 
              V V  B      T ,{  }   i          V T      B ,{  i }
                               i




O. G. Benvenuto and G. Lugones, Phys. Rev. D 51, 1989 (1995)
    G. Lugones and O. G. Benvenuto, ibid. 52, 1276 (1995)
C.   m  m(  B )
                B                                          T 
        p                              ,        i  i                        ,
              V V  B       T ,{  i }
                                                 V   i          V T    B ,{  i }


                         G. X. Peng, et. al. Phys. Rev. C 59, 3452 (1999)
                        G. X. Peng, et. al. Phys. Rev. C 62, 025801(2000)
     m  m(  B , T )
                    ~             ~
             ~                  m * i
       p    V      B                ,
                  V        i m i   B
                                   *

                      ~       ~                  ~
           ~                                 m * i
            i      T             T               ,
               i     i    T  ,{  }      i m i T
                                                  *
                                            B    i



                        X. J. Wen, et. al. Phys. Rev. C 72, 015204 (2005)
 Inconsistency of Traditional Thermodynamic
      Treatments with Partial Derivative
 Differential relation for reversible process
 Ω = Ω(T, V, μ).




 If m*=m*(T,ρ), Ω = Ω(T, V, μ, m*(T,ρ), ), the
  Massieu’s Theorem breaks down.
Quasi-particle approximation
     H eff    (k , T ,  )a ak
                               
                               k
              k

    ( k , T ,  )  k  m (T ,  )
                     2     2


 G  U  TS  pV  F    0; (   0)
           S dS dp
                     ;
           V dV dT
          U   dp(T )
         T         p(T ).
          V    dT
 Thermodynamic inconsistency
           For QMDD Model




 ρ=N/V →μ, fixed {T,μ} equals fixed {T,ρ}
 Change V, N must change, too.
                  f      f   f   y 
                        
 According to     x  g   x  y  y  x  x  g
                                                       , we
 write down the invariables explicitly
=0
Inconsistent with
 Reversible Process           fix t
  equilibrium state
 Suppose T=T0, ρ=ρ0,
       m*(T, ρ)=m*(T0,ρ0)
 All formulae in equilibrium state are
  applicable
Thermodynamic Consistent Treatment
 In equilibrium state
Calculation of U from the definition
Consist with the interaction-free
quasi-particle picture
Calculation of S from the definition
Calculation of S from partial derivative
 Our treatment can be expressed by
 considering the quasi-particle mass as
 independent variable
 Ordinary thermodynamic variables depend
  on the collection of the subsystem only.
 Mass is an intrinsic quantity of a particle, it
  does not affect on collective thermodynamic
  properties.
 Effective mass m*(T, ρ) includes dynamic
  interaction, confinement mechanism, etc.
 But the macro thermodynamic variables
  cannot describe these micro dynamic
  interactions. We must choose new
  variables to represent these dynamic
  interactions or the medium effect.
 Introducing m* in quasiparticle physical
  picture to represent the medium effect and
  taking it as a variable is a twin in
  thermodynamics of quasiparticle system.
QMDD model
Old treatment




Our treatment
Our treatment




Old treatment I




Old treatment II
         Contribution of Vacuum
 Within the statistical frame, the pressure is
  positive definite, p=-Ω/V>0
 In MIT bag model, B0 is added to energy while
  subtracted in pressure as vacuum contribution,
  negative pressure can be realized
    Constraint on Vacuum




Ω0(ρB) can be obtained by integration
               Conclusion
 For model Hamiltonian with effective mass
 quasiparticles, an intrinsic degree of
 freedom m* must be introduced
     d   SdT  pdV   N i di  Xdm * .
                          i


 All ambiguities are solved
 Correct physical picture after the vacuum is
 introduced
           PRC Referee’s Report
 This is an interesting paper which should be published in
  PRC. The authors explain the inconsistencies in previous
  thermodynamical treatments of quark matter within the
  quark mass density-dependent model and show how the
  model can be used self-consistently by introducing the
  quasiparticle mass as a new independent variable. This
  leads to reasonable numerical results resembling those
  obtained with the MIT bag model, but more importantly it
  leads to an improved understanding of the physics.

 In fact as the authors mention in the paper their method
  may be more widely applicable to other systems where
  medium effects can be described by an effective mass,
  and my only suggestion for changes in the manuscript is to
  include this statement in the Abstract in order to attract
  more readers from other subfields.
Thank you!