First order dissipative hydrodynamics from an effective covariant kinetic theory
The first order hydrodynamic evolution equations for the shear stress tensor, the bulk viscous pressure and the charge current have been studied for a system of quarks and gluons, with a non-vanishing quark chemical potential and finite quark mass. The first order transport coefficients have been obtained by solving an effective Boltzmann equation for the grand-canonical ensemble of quasiquarks and quasigluons. We adopted temperature dependent effective fugacity for the quasiparticles to encode the hot QCD medium effects. The non-trivial energy dispersion of the quasiparticles induce mean field contributions to the transport coefficients whose origin could be directly related to the realization of conservation laws from the effective kinetic theory. Further, the relative significance of dissipative quantities has been investigated through their respective ratios. Both the QCD equation of state and chemical potential are seen to have a significant impact on the QGP evolution. The first order viscous corrections to the time evolution of temperature along with the description of pressure anisotropy of the system have also been explored.
High energy heavy-ion collision (HIC) experiments in Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) have realized the existence of a new state of matter-quark gluon plasma (QGP), as a near-perfect fluid STAR (); Aamodt:2010pb (); Heinz:2008tv (). Relativistic hydrodynamics has been successfully employed to describe the space-time evolution of the created deconfined nuclear matter; see Refs. Jeon:2015dfa (); Florkowski:2017olj (); Heinz:2013th (); Braun-Munzinger:2015hba (); Jaiswal:2016hex () for recent reviews. On the other hand, other input parameters such as equation of state and transport coefficients, have been estimated from the microscopic theories. The inclusion of dissipative effects in the QGP evolution is significant for explaining the quantitative behavior of experimental observables in the HIC, i.e., collective flow, transverse momentum spectra etc. Denicol:2010xn (); Baier:2006um (); Baier:2007ix (); Denicol:2012cn (); Bhalerao:2013pza (). The theoretical explanation of the hadron elliptic flow in the RHIC with dissipative hydrodynamic evolution provide the evidence of transport processes in the QCD medium Luzum:2008cw (). The relevance of the transport process in the HIC is reconfirmed in ALICE:2016kpq (); Adam:2016izf (); Abelev:2013cva (); Adam:2016nfo ().
There have been various approaches/attempts for the estimation of the transport parameters of the hot QCD medium Deb:2016myz (); Ghosh:2015mda (); Mitra:2016zdw (); Jaiswal:2014isa (); Florkowski:2015lra (). To explore the relative significance of transport parameters, their ratios have been studied in recent literature Marty:2013ita (); Mitra:2017sjo (). The quantitative estimation of shear viscosity from experiments has been widely investigated in several works Niemi:2011ix (); Niemi:2012ry (); Romatschke:2007mq (); Song:2008hj (); Song:2010aq (); Song:2011qa (); Song:2011hk (); Schenke:2010rr (); Gale:2012rq (). In parallel, there have been some attempts to study the effect of bulk viscosity in the evolution of the QGP Ryu:2015vwa (); Huang:2010sa (); Denicol:2014vaa (); Dobado:2011qu (); Bluhm:2010qf (). Notably, the effect of dissipative charge current has received less attention compared to the viscous coefficients in the framework of dissipative hydrodynamics. This can be attributed to the fact that the net baryon number and chemical potential are insignificant in the very high energetic collisions. However, for the lower collision energies probed in the RHIC low-energy scan and for upcoming experiments at Facility for Antiproton and Ion Research (FAIR), the baryon chemical potential can no longer be neglected. In addition to the effects of the chemical potential, the finite quark mass corrections are also significant in the evolution of the QGP in this context. This sets the motivation to investigate the hydrodynamic evolution of the QGP with a non-vanishing baryon chemical potential and finite quark mass.
The description of the QCD medium evolution requires the knowledge of microscopic description of thermodynamic quantities of the medium along with the appropriate momentum distribution functions of its effective degrees of freedom (effective quarks/antiquarks and effective gluons). To that end, encoding the thermal medium effects encoded in the hot QCD equations of state (computed within lattice QCD or Hard Thermal Loop effective theory) in terms of effective quasiparton degrees of freedom with nontrivial dispersion relations has turned out to be a viable approach. The quasiparticle description of thermodynamic and transport properties of the hot QCD/QGP medium have been investigated in several works Mitra:2016zdw (); Mitra:2017sjo (); Kurian:2018dbn (); Kurian:2017yxj (); Kurian:2018qwb (); Tinti:2016bav (); Alqahtani:2016rth (); Alqahtani:2017jwl (); Rozynek:2018tev (). In the current analysis, we utilize the effective fugacity quasiparticle model (EQPM) Chandra:2011en (); Chandra:2007ca () for the effective description of the QGP. The microscopic framework for the estimation of transport coefficients in the current analysis is done within the covariant kinetic theory with proper collision kernel. We employ the Chapman-Enskog method within the relaxation time approximation (RTA) to solve the relativistic transport theory. The mean field term in the effective covariant kinetic theory with the EQPM can be realized from the conservation laws as described in the Ref. Mitra:2018akk ().
The goal of the current analysis is to investigate the mean field corrections to the dissipative quantities with non-vanishing baryon chemical potential and quark mass, within an effective quasiparticle model. The relative behavior of different dissipative processes can be estimated with their respective ratios in the light of the mean field contributions and finite quark chemical potential. We study the viscous corrections to the time evolution of temperature and pressure anisotropy by analyzing the boost invariant longitudinal expansion. These aspects are crucial in the investigation of the hydrodynamic evolution of the QGP from the covariant effective kinetic theory.
The manuscript is organized as follows. The mathematical formulation of the first order dissipative hydrodynamic evolution equations from the EQPM covariant kinetic theory along with the description of longitudinal Bjorken flow is presented in section II. Section III deals with the discussions on the mean field contributions and the relative significance of transport coefficients. Finally, in section IV, the conclusion and outlook have been presented.
The formalism for the estimation of dissipative hydrodynamic evolution of the QGP consists of the quasiparticle modeling followed by the setting up of the effective covariant kinetic theory of the system away from equilibrium. The current analysis is based on a covariant kinetic theory for hot QCD medium recently developed by Chandra and Mitra Mitra:2018akk () employing the effective fugacity quasiparticle model (EQPM) Chandra:2011en (); Chandra:2007ca (). Here, we have extended the approach to investigate the transport properties of the hot QCD systems with finite quark chemical potential and quark-antiquark masses. There are several other quasiparticle models present in the literature to describe hot QCD medium which include models with effective masses for the quasipartons kamf (); Peshier:1995ty (); DElia:1997sdk (); DElia:2002hkf (); Castorina:2007qv (); Castorina:2005wi (), a self-consistent, single parameter quasiparticle models with temperature dependent effective mass Bannur:2006js (); Koothottil:2018akg (), NJL and PNJL based quasiparticle models Dumitru:2001xa (); Fukushima:2003fw (); Ghosh:2006qh (), and recently proposed quasiparticle model based on the Gribov-Zwanziger quantization Su:2014rma (); Florkowski:2015dmm (); Bandyopadhyay:2015wua ().
ii.1 QCD thermodynamics and the effective covariant kinetic theory with finite chemical potential
Realizing the hot QCD as a Grand-canonical ensemble, the EQPM interprets the hot QCD equation of states (EoS) as the non-interacting quasipartons with effective fugacities. The equation of states (EoSs) has been carefully embedded in the quasigluon and quasiquark/antiquark effective fugacities and respectively, for both isotropic and anisotropic QCD medium. Here, we considered the flavor lattice QCD EoS for the effective description of QGP Cheng:2007jq (); Borsanyi:2013bia ().
The quasiparticle energy-momentum tensor can be defined in terms of dressed momenta within EQPM as follows Mitra:2018akk (),
where and is the invariant phase space factor. We are considering nonzero masses for quarks (with MeV, MeV and MeV for up, down and strange quarks respectively) and for quarks/antiquarks whereas for gluons . Here, is the degeneracy factor of the th species and is defined as the projection operator orthogonal to fluid velocity with the metric diag . The EQPM parton distribution function in the local rest frame is defined as,
where . The dispersion relation encodes the collective excitation of quasiparton, relates the quasiparticle and bare four-momenta as follows,
In the local rest frame, the zeroth component of the four-momenta gets modified as,
The extension of the EQPM to finite quark chemical potential ( is quite straightforward. The quasiquark phase-space momentum distribution becomes,
Since the effective fugacities , are not related with any conserved number current in the QGP medium, the temperature dependence of , remain intact so that one can get the correct limit in the case of vanishing chemical potential. At , . It is important to note that varies with , where GeV is the QCD transition temperature for the current analysis. Next, we focus on the particle number flow . The quasiparticle description of the flow in terms of dressed momenta takes the following form,
where is the irreducible tensor of rank one. The relevant thermodynamic quantities such as energy density, pressure, the speed of sound and number density can be obtained following their basic thermodynamic definitions.
From Eq. (II.1), we obtain the expression of the energy density and the pressure , respectively, within the EQPM by using the following definitions
For the case of finite quark mass and non-vanishing baryon chemical potential , the quark contribution to the energy density and pressure can be expressed in terms of modified Bessel function of second kind, , as
where and . Similarly, one can obtain the quasiparticle net number density using the definition . The number density is then given by
One can see that in the and limit, the above expressions for energy density and pressure reduce to that obtained in Ref. Florkowski:2015lra ().
In the massless case, the EQPM energy density and the pressure of the hot QGP with a non-zero can be obtained in terms of functions and have the following form,
Similarly, the net number density in the massless limit takes the following form,
One can see that in the limit , the above expression reduces to that obtained in Ref. Jaiswal:2015mxa ()
The macroscopic definition of viscous tensor and the particle diffusion current requires the non-equilibrium or collisional part of the distribution function of the particles. For the system close to local thermodynamic equilibrium, the non-equilibrium quasiparton phase space distribution function takes the form , where . Macroscopically, the energy-momentum tensor in the non-equilibrium case can be decomposed as,
where and are the shear tensor and bulk viscous pressure respectively. Similarly, the particle four-current can be macroscopically described as,
Note that the above expressions for energy-momentum tensor and particle four-current are written for fluid four-velocity defined in Landau frame.
The projection of and conservation equations along and orthogonal to gives,
where and , with being the traceless symmetric projection operator orthogonal to the fluid velocity . Since the energy density and pressure can be expressed as a function of temperature, we can further rewrite the Eqs. (16), (17) and (18) as,
where the squared velocity of sound at constant , where is the entropy density of the medium Haque:2014rua ().
The shear stress tensor can be expressed in terms of within EQPM as follows Mitra:2018akk (),
Similarly, the bulk viscous pressure and the particle diffusion current can be defined respectively as follows,
We will use the above equations for disipative quantities to obtain their first-order expressions and corresponding transport coefficients.
The relativistic transport equation quantifies the rate of change of quasiparton phase space distribution function in terms of collision integral and has the following form,
in which is the force term defined from the conservation of energy momentum and particle flow. In the current EQPM framework, the collision integrals is defined in the relaxation time approximation (RTA), where the thermal relaxation linearizes the collision term as,
To obtain , we solve the relativistic Boltzmann equation with RTA using the Chapman-Enskog (CE) expansion.
ii.2 First order dissipative evolution equation
The first order correction to distribution functions for quarks, anti-quarks and gluons can be obtained from the Boltzmann equation Eq. (24) by considering an iterative Chapman-Enskog like solution Jaiswal:2013npa (); Jaiswal:2013vta (). For our current effective kinetic theory, we obtain the following form,
where and .
With the definitions of the in Eqs. (26), (27) and (28), we can obtain the first order evolution equation for dissipative quantities. Assuming the thermal relaxation time to be independent of four-momenta and keeping terms up to first-order in gradients, we obtain
where the coefficients have the following form,
Various thermodynamic integrals labeled by and , appearing in the above expressions, are defined as,
where is the distribution part of the integrand.
we can obtain the coefficients of bulk viscosity, shear viscosity and charge conductivity as , and , respectively. Note that we consider a special case where the relaxation times for all particle species are same. The general case with different thermal relaxation time is beyond the scope of the current analysis.
For the case of massive quasipartons, the scalar thermodynamic integrals and can be expressed in terms of the modified Bessel function of second kind
where the function is defined as,
The form of above integrals for the massless case, in terms of functions, is presented in the appendix A.
ii.3 Longitudinal boost-invariant expansion
To model the dissipative hydrodynamical evolution of the QGP formed in the heavy-ion collision experiments, we employ the Bjorken’s prescription Bjorken:1982qr () for one-dimensional boost invariant expansion. Here, we consider the case of vanishing baryon chemical potential. The evolution equation of the energy density for purely longitudinal boost-invariant expansion can be expressed in terms of Milne coordinates , where and resulting in with the metric tensor given by Tinti:2016bav (). Employing the Milne coordinate system, the energy evolution equation Eq. (16) gets simplified to,
where we have used , and . We numerically solve Eq. (50) to study the evolution of viscous nuclear matter with the values of dissipative quantities given in Eq. (32) and Eq. (33), imposing the LEoS. The initial condition in RHIC (for Pb-Pb collision) is GeV at fm and in LHC (for Au-Au collision) is GeV at fm El:2007vg (). We estimated the temperature evolution by assuming relaxation time to be same for both bulk and shear parts ( fm). With this conditions, we can investigate the proper time evolution of longitudinal pressure () to transverse pressure (), , where is the equilibrium thermodynamic pressure.
Iii Results and discussions
We initiate the discussion with the temperature dependence of mean field corrections to the shear tensor, bulk viscous pressure and the particle diffusion, respectively, of the hot QGP with finite quark chemical potential. The mean field force term from the effective theory appears as the mean field corrections to the transport coefficients of the system. As mentioned earlier, the force term consists of the modified part of the EQPM dispersion relation . At high temperature region , the fugacity parameters are the slowly varying functions of temperature. Since is the temperature gradient of the , the mean field effects are not significant at higher temperature region. The mean field contributions to first order coefficients of the shear tensor and bulk viscous pressure at quark chemical potential GeV are depicted in Fig. 1 (left panel).
To quantify the mean field corrections, we choose the appropriate gluon and quark degeneracy factors respectively as and , where is the number of flavors, is the spin degrees of freedom and is the number of colors. Since the mean field corrections at high temperature regimes are negligible, the ratio asymptotically tends to unity. However, the mean field contributions due to the quasiparticle excitation are significant in the lower temperature regime. The effects of mean field contributions to the dissipative quantities are shown in Fig. 1. We observe the quantitative difference in the and with and without the mean field corrections at low temperature regimes. In Fig. 1 (right panel), the mean field effects to the first order coefficient of particle diffusion is shown for different quark chemical potential . The dependence of finite quark mass and baryon chemical potential to the mean field contributions are separately shown in the left and right panel of Fig. 1 respectively. The mean field correction to the transport parameters with binary, elastic collisions at and is described in Ref. Mitra:2018akk (). The effects of quark mass and chemical potential are visible in the low temperature regimes whereas in the higher temperature regimes the mean field contributions are almost independent on and .
In Fig. 2, we show the temperature dependence of the ratio of the coefficient of the bulk viscous tensor to that of the shear tensor at GeV. In the RTA, the ratio becomes , where is the bulk viscosity of the hot QGP medium. Within the EQPM, the term is nonzero even for the massless case. The squared speed of sound tends to the Boltzmann limit at very high temperature. Since the bulk viscosity is proportional to this term, the effective description of with EQPM tends to zero at high temperature regime. We observe that the temperature behavior of the ratio has a decreasing trend with the increase in temperature. We observe that the quark mass correction and mean field corrections are more visible in the low temperature regime near to the transition temperature . Further, we compared the results with other parallel work and the lattice results. We found that our observations are consistent with the results of Bluhm:2011xu (); Meyer:2007ic (); Meyer:2007dy ().
The relative significance of charge conductivity and shear viscosity could be understood in terms of the ratio . Within RTA, the quantity . The temperature behavior of the ratio is plotted in Fig. 3 (left panel) for different quark chemical potential. The ratio becomes almost constant at high temperature regions, whereas it drops for the low temperature regime, indicating that the conductivity of the medium is relatively small to the shear viscosity in the regime as compared to very high temperature regimes. We observe a similar trend in the temperature behavior of the dimensionless ratio , in which is the coefficient of thermal conductivity. The quantity is defined as within RTA Jaiswal:2015mxa (). In the high temperature limit, the ratio reduces to as shown in the Fig.3 (right panel). For the non-interacting QGP , the value of the constant becomes Jaiswal:2015mxa (). However, it should be noted that for the realistic EoS, the value of is equal to . The EoS dependence of the viscous coefficients and conductivity are entering through the effective quasiparton fugacities. Also, the mean field effects to the ratio are more visible in lower temperature regime whereas the effects are negligible at higher temperature limit. In the ideal limit , the EQPM results can be reduced to the first order dissipative hydrodynamic evolution equation with appropriate coefficients as described in Jaiswal:2015mxa ().
In Fig.4 (left panel), we depicted the proper time evolution of temperature and pressure anisotropy in ideal and first order hydrodynamics with initial temperature MeV at proper time fm/c. We assumed the Navier-Stokes initial condition for shear and bulk viscous part respectively as and , with the thermal relaxation time fm. The temperature evolution based on the first order dissipative hydrodynamics shows slower temperature drop with proper time compared to the ideal evolution. Following the temperature evolution, the proper time dependence of the pressure anisotropy is shown in Fig.4 (right panel). We see that compared to the first order evolution results for in case of non-interacting Boltzmann particles Jaiswal:2013vta (), there is a slightly faster approach to isotropization in the present EQPM model.
Iv Conclusion and Outlook
In this paper, we have derived the first order dissipative hydrodynamic evolution equations within an effective covariant kinetic theory by realizing the system as a grand canonical ensemble of gluons and quarks, with a finite baryon chemical potential and non-zero quark mass . The covariant effective kinetic theory is employed for the hot QCD matter within the EQPM. The thermal medium effects have been encoded through the EQPM by introducing the lattice equation of state in phase space momentum distribution through the effective fugacity parameter. We observed that the mean field contributions that emerge from the covariant kinetic theory induce sizable modification to the first order coefficients of the shear stress tensor, bulk viscous pressure and the particle diffusion of the hot QGP medium in the temperature regime near to . However, the modifications to the first order coefficients are negligible at higher temperatures . In the massless limit, our estimations at agree with results of Ref. Mitra:2018akk (), for the binary elastic collisions.
We further studied the ratio of viscous coefficients and compared the results with other parallel works. Furthermore, the relative significance of the charge conductivity and thermal conductivity with the viscous shear tensor have been investigated by evaluating the ratio and respectively within RTA for different quark chemical potential. We found that at the lower temperature the charge conductivity is relatively smaller compared to the results at higher temperatures. Also, the effect of the baryon chemical potential is more visible in the temperature regime near to . The proper time evolution of temperature and pressure anisotropy are seen to be sensitive to the viscous effects and the equation of state. Finally, various predictions of the current work, turned out to be consistent with the other parallel results.
The analysis presented in the manuscript is the first step towards the higher order (second and third) dissipative hydrodynamic evolution equation from the effective covariant kinetic theory within the EQPM. The investigation of the hydrodynamic evolution equations for the hot magnetized QGP medium (magnetohydrodynamics) would be another interesting direction to pursue. In addition, deriving the transport coefficients using a more realistic collision term is another problem worth investigating. We leave these problems for future work.
M. K. would like to acknowledge the hospitality of NISER Bhubaneswar. V. C. would like to acknowledge SERB for the Early Career Research Award (ECRA/2016), and Department of Science and Technology (DST), Govt. of India for INSPIRE-Faculty Fellowship (IFA-13/PH-55). A. J. is supported in part by the DST-INSPIRE faculty award under Grant No. DST/INSPIRE/04/2017/000038. We are indebted to the people of India for their generous support for research in basic sciences.