A Consistent Thermodynamic Treatment for Quark Mass Density-Dependent Model Ru-Keng Su Physics Department Fudan University

Difficulties  In relativistic energy dispersion relation:  Ω becomes an explicit function of m:  How will the thermodynamic formulae with the partial derivatives become?

Different treatments with extra terms from partial derivatives A.

O. G. Benvenuto and G. Lugones, Phys. Rev. D 51, 1989 (1995) G. Lugones and O. G. Benvenuto, ibid. 52, 1276 (1995) B.

X. J. Wen, et. al. Phys. Rev. C 72, (2005) G. X. Peng, et. al. Phys. Rev. C 59, 3452 (1999) G. X. Peng, et. al. Phys. Rev. C 62, (2000) C.

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

 Thermodynamic inconsistency

For QMDD Model  ρ=N/V →μ, fixed {T,μ} equals fixed {T,ρ}  Change V, N must change, too.

 According to, we write down the invariables explicitly

=0

Inconsistent with

 Reversible Process fix t equilibrium state  Suppose T=T 0, ρ=ρ 0, m*(T, ρ)=m*(T 0,ρ 0 )  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

Our treatment Old 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, B 0 is added to energy while subtracted in pressure as vacuum contribution, negative pressure can be realized

Constraint on Vacuum Ω 0 (ρ B ) can be obtained by integration

 For model Hamiltonian with effective mass quasiparticles, an intrinsic degree of freedom m* must be introduced  All ambiguities are solved  Correct physical picture after the vacuum is introduced Conclusion

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!