Research Paper Notes on Equation of State

Research Paper Notes on Equation of State

List of papers

 * arXiv:nucl-th/0702030 Temperature dependence of sound velocity and hydrodynamics of ultra-relativistic heavy-ion collisions
 * arXiv:0801.4361 Initial condition for hydrodynamics, partonic free streaming, and the uniform description of soft observables at RHIC
 * arXiv:nucl-th/0505036 Anisotropy of ﬂow and the order of phase transition in relativistic heavy ion collisions
 * arXiv:0912.2541 QCD Equation of State and Hadron Resonance Gas
 * arXiv:1106.6227 Equation of state at finite baryon density based on lattice QCD
 * arXiv:1204.6710 QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order mu2
 * arXiv:0712.0947 Soft heavy-ion physis from hydrodynamics with statistical hadronization predictions for the Large Hadron Collider
 * arXiv:0805.1756 Multiplicity scaling in ideal and viscous hydrodynamics
 * arXiv:hep-ph/0607079 Entropy of expanding QCD matter
 * arXiv:0903.4379 Equation of state and QCD transition at ﬁnite temperature
 * arXiv:1007.2580 The QCD equation of state with dynamical quarks
 * arXiv:hep-ph/0603048 Quark mass thresholds in QCD thermodynamics
 * arXiv:hep-ph/0510096 3D relativistic hydrodynamic computations using lattice-QCD-inspired equations of state
 * arXiv:0909.0391 Nearly perfect ﬂuid in Au+Au collisions at RHIC
 * arXiv:0905.3099 Hydrodynamics with a chiral hadronic equation of state including quark degrees of freedom
 * Phys.Rev. D47 (1993) 2068 The Phase structure of strange matter

arXiv:nucl-th/0702030 Temperature dependence of sound velocity and hydrodynamics of ultra-relativistic heavy-ion collisions
This the paper by Florkowski (sent to Danuce). The EoS is based on the lattice data and hadronic resonance gas model. The main point of the work is how to interpolate the sound velocity in the transition region. For a strict first order phase transition, the sound velocity is zero in the transition region (since the volume, therefore the energy density, changes while the pressure remain the same), therefore, in the past, the authors believe there is a minimum of sound velocity vs. T at the transition point. The $$c_s^2$$ vs. $$T$$ curve is subjected to the calculated entropy by lattice QCD owing to thermodynamical relations. However, the hydrodynamical evolution says elsewise since the resulting evolution time is not consistent with the HBT radius, i.e., the observed radius is too small comparing to the large calculated evolution time. The hydrodynamical evoltuion is of 1+1 (boost invariant + cylindrical symmetry), and the EoS considers only zero bayron density.

arXiv:0801.4361 Initial condition for hydrodynamics, partonic free streaming, and the uniform description of soft observables at RHIC
This is a general purpose work in hydrodynamics published on PRL. The study mainly focus on the effect of Glauber type initial conditions on the resulting HBT radii. The calculations were carried out by a 2+1 code, their EoS considers lattice data and hadron resonance gas model with zero baryon density. As a matter of fact, the paper itself did hardly mention the EoS. It is the same EoS introduced in arXiv:nucl-th/0702030.

arXiv:nucl-th/0505036 Anisotropy of ﬂow and the order of phase transition in relativistic heavy ion collisions
This is the first (well known) paper by Huovinen which discusses the EoS based on lattice data. The author claimed that before the work, all published calculations were done using EoS of the first order phase transition. Surprisingly, the author concludes that the lattice based EoS gives poor estimation of v2, which is as bad as hadronic EoS. In fact, as pointed out explicitly in a subsequent work, the main discrepancies only come from the proton flow. The author used a lattice inspired EoS by Schneider and Weise with zero bayron density and it is connected to hadronic resonance gas model by parameterization. The work also compared other EoS, including EoS of the first order phase transition. It used a full 2+1 hydro code. Another important result is that, the author estimated that finite chemical potential gives an effect of less than 2%.

arXiv:0912.2541 QCD Equation of State and Hadron Resonance Gas
This is the most cited work by Huovinen et.al. The paper addresses the discrepancies between hadronic resonance gas model and lattice results at low temperature (~180 MeV). The discrepancies come from, as the authors argued, the difference between the physical mass and lattice mass of hadron resonances. Since the resonance gas model describe well experimental observations, the deviations were therefore attributed to the continuum limit (finite lattice spacing) in lattice calculations. A new EoS parameterization was proposed by the authors, the model focus on the trace anomaly $$\Theta \equiv {T^\mu}_\mu = e - 3P$$. The EoS adopts the lattice data at high temperature where the calculations are reliable and it is connected smoothly to the hadron resonance gas model at low temperature by requiring that the trace anomaly as well as its first and second derivatives are continuous. As a result, the parameterization describes the lattice at high temperature, while in the low temperature region, it stays above the lattice results for the reason above-mentioned. The work also compared and discussed different EoS, the comparison to the data was quite limited.

arXiv:1106.6227 Equation of state at finite baryon density based on lattice QCD
The authors used Taylor expansion coefficients obtained from lattice calculation to extend their previous EoS to finite baryon density. Again, the parameterization is required to match the hadron resonance gas model at low temperature TSW with continuous Taylor coefficient as well as its first and second derivatives.

arXiv:1204.6710 QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order mu2
Lattice有很多实验组.有两个可能是最重要的,一个是采用stout作用量的Wuppertal-Budapest合作组,另一个是采用HISQ作用量的HotQCD合作组.这个工作是采用stout作用量的结果.

(2.5) 这个式子是由化学势平衡条件加上奇异夸克化学势$$\mu_S=0$$为零得到的.即对任何化学平衡的粒子粒子$$a$$,有$$\mu_a=N^B_a\mu_B+N^S_a\mu_S+N^E_a\mu_E$$.其中$$a$$可以是$$u,d,s$$夸克,而$$B,S,E$$为所有问题中涉及到的守恒荷,比如重子数奇异数电荷数.我们用大写字母来表达某守恒荷,小写字母来表达某粒子.而比如$$N^E_a$$代表$$a$$粒子携带的电荷$$E$$数目.这个式子的推导见本笔记下面的具体讨论.注意上述条件并不意味着奇异数密度为零,也不意味着奇异夸克数密度为零.

(2.6) 这里的第三个式子是有奇异夸克数密度为零的条件,即$$n_s=\frac{1}{\beta V}\frac{\partial ln\Omega}{\partial \mu_s}$$.注意,这同时意味着奇异数密度为零$$0=n_S=\frac{1}{\beta V}\frac{\partial ln\Omega}{\partial \mu_S}$$.将展开(2.3)代入上面表达式,即得到轻夸克和奇异夸克化学势之间的关系.注意这个结果与理想气体模型并不冲突,因为对后者按定义(2.4),$$\chi_2^{us}=0$$,这样奇异夸克化学势为零$$\mu_s=0$$.这和前面(2.5)中的条件是很不一样的.

arXiv:0712.0947 Soft heavy-ion physis from hydrodynamics with statistical hadronization predictions for the Large Hadron Collider
The work is devoted to a general purpose study of heavy-ion collisions using hydrodynamic model. Its EoS is based on sound velocity, it reproduced the sound velocity of lattice data at high temperature and that of pion at low temperature, a simple smooth interpolation was intuitively introduced in the between. It deals with zero baryon density. The hydro code was 2+1.

arXiv:0805.1756 Multiplicity scaling in ideal and viscous hydrodynamics
Another general purpose work in hydrodynamics by Song and Heinz. The focus of the study is the viscous effect in hydrodymics. The authors discussed several different EoS in the work. The EoS in focus considers both lattice data and hadron resonance gas model, (in instead of considering sound velocity or trace anomaly) the interpolation was carried out in terms of $$P(e)$$ curve, and sound velocity was evaluated afterwards. As the authors pointed out themselves, their treatment was not fully thermodynamically consistent. The hydro code was 2+1.

arXiv:hep-ph/0607079 Entropy of expanding QCD matter
This work involves an analysis of effective number of degree of freedom during the deconfinement phase transition. The work focuses to understand whether the decrease of effective number of quasiparticles during the transition is in contradict to the value of the entropy of the system. The calculation only involves a fit to the lattice data, and the hydro evolution is 1+1.

arXiv:0903.4379 Equation of state and QCD transition at ﬁnite temperature
An QCD EoS obtained by lattice calculations. Calculations were performed by using two different improved staggered Fermion actions, namely, asqtad and q4 actions.

arXiv:1007.2580 The QCD equation of state with dynamical quarks
An QCD EoS obtained by 2+1 staggered ﬂavors and one-link stout.

arXiv:hep-ph/0603048 Quark mass thresholds in QCD thermodynamics
In this work, the authors calculated an EoS using perturbative QCD, where finite quark mass was taken into consideration. The results are compared with lattice calculations.

arXiv:hep-ph/0510096 3D relativistic hydrodynamic computations using lattice-QCD-inspired equations of state
In this work, the ideal quark gluon gas model is connected to the hadron resonance gas model by a parameterization where a phenomenological critical point was introduced. Below the critical point, the transition is of a smooth cross-over, beyond the critical point, the transition of first order. The advantage of this phenomenological parameterization is that it naturally incorporates finite baryon density. Unfortunately, the model does not incorporate information from lattice calculations.

arXiv:0909.0391 Nearly perfect ﬂuid in Au+Au collisions at RHIC
An EoS is proposed in this work which consists of a intuitive parameterization to smoothly connect lattice data and hadron resonance gas. The EoS considers zero baryon density. The hydrodynamic code is 2+1 viscous.

arXiv:0905.3099 Hydrodynamics with a chiral hadronic equation of state including quark degrees of freedom
An EoS is proposed which incorporate a chiral hadronic SU(3) model, it is similar to the PNJL model which contains a critical end point. The disadvantage of the model is, of course, that it does not include all observed degree of freedom in the hadronic phase, and it does not match perfectly to the lattice data at high temperature. The hydrodynamical calculations involve UrQMD initial conditions and final hadron cascade, the hydrodynamical evolution is ideal 3+1.

Phys.Rev. D47 (1993) 2068 The Phase structure of strange matter
化學平衡導致的化學勢關係以及推導.

考慮強子物質和夸克物資達到化學平衡,

首先考慮由質子組成的強子物質與夸克物資平衡.即


 * $$p \leftrightarrow 2u + d$$

粒子數守恆意味著,


 * $$dn_u =2dn_p$$
 * $$dn_d =dn_p$$

而化學平衡意味著


 * $$\mu_u dn_u+\mu_d dn_d=\mu_p dn_p$$

從而得到


 * $$2\mu_u+\mu_d=\mu_p$$

接著考慮由質子和中子組成的強子物質與夸克物資平衡.即


 * $$p \leftrightarrow 2u + d$$
 * $$n \leftrightarrow u + 2d$$

粒子數守恆意味著,


 * $$dn_u =2dn_p+dn_n$$
 * $$dn_d =dn_p+2dn_n$$

而化學平衡意味著


 * $$\mu_u dn_u+\mu_d dn_d=\mu_p dn_p+\mu_n dn_n$$

從而得到


 * $$2\mu_u+\mu_d=\mu_p$$
 * $$\mu_u+2\mu_d=\mu_n$$

容易看到增加其他強制物資不會改變上面已經得到的化學勢關係.實際上不難發現,不論是重子還是夸克$$A$$,其化學勢都可以方便的寫成


 * $$\mu_A=n^B_A\mu_B+n^S_A\mu_S$$

其中$$n^B_A$$是該粒子帶有的重子數數目,以此類推.如果再進一步考慮反粒子,我們就得到方程(7).