Shear viscosity of an ultrarelativistic Boltzmann gas with isotropic inelastic scattering processes
We derive an analytic expression for the shear viscosity of an ultra-relativistic gas in presence of both elastic and inelastic processes with isotropic differential cross sections. The derivation is based on the entropy principle and Grad’s approximation for the off-equilibrium distribution function. The obtained formula relates the shear viscosity coefficient to the total cross sections and of the elastic resp. inelastic processes. The values of shear viscosity extracted using the Green-Kubo formula from kinetic transport calculations are shown to be in excellent agreement with the analytic results which demonstrates the validity of the derived formula.
pacs:47.75.+f, 24.10.Lx, 12.38.Mh, 25.75.-q
Dynamics of the hot and dense state of nuclear matter, produced in heavy-ion collisions at modern accelerator facilities like RHIC at Brookhaven National Laboratory or LHC at CERN, can be described, with exception of the very early times governed by off-shell dynamics, by means of kinetic transport theory or relativistic hydrodynamics. In fact, these two approaches have been widely used for research of the properties of the so-called Quark-Gluon Plasma (QGP), early-time dynamics of which determines the properties of hadronic and electromagnetic secondaries measured experimentally hydros (); Xu:2004mz (); Xu:2007jv (). Whereas the kinetic transport theory is based on a microscopic paradigm and describes evolution of the phase-space distribution function by solving the corresponding transport equations, hydrodynamics is a macroscopic theory describing space and time evolution of macroscopic fields like the energy-momentum tensor and particle number four-vector. For the hydrodynamic theory the transport coefficients are an extrinsic input and must be calculated from the underlying field theory, which for QGP is the Quantum Chromodynamics (QCD). Calculations of the transport coefficients are based on the formal correspondence between the kinetic theory and the hydrodynamics, which can be obtained by integration of the microscopic equations over the phase space. For example, such correspondence is discussed in IS (); Muronga:2003ta (); Baier:2006gy (); Betz:2009zz (); Denicol:2010xn (). Moreover, kinetic transport calculations have often been used as a benchmark for hydrodynamic models, as for instance in Refs. Huovinen:2008te (); El:2009vj (); Bouras:2010hm (), which again requires a consistent connection between the cross section on the one hand and transport coefficients on the other.
For a strongly interacting QGP, the shear viscosity coefficient has been calculated in the perturbative regime as function of the coupling parameter for instance in Refs. Arnold:2000dr (), El:2008yy (); Wesp:2011yy (); Reining:2011xn (). Such calculations account for the radiative processes occurring in the QGP, which have an important contribution to the isotropization of the medium. The role of inelastic processes was understood by means of the transport cross section or transport rate Xu:2007aa (), which are analogous to the total cross section and collision rate but are additionally weighted by the angle of outgoing particles. In contrast, calculations reported in Refs. Chen:2010xk () do not see a strong contributions of the inelastic processes. In Ref. Fuini:2010xz () the shear viscosity of a QGP was calculated employing two-particle scatterings, i.e. without taking into account the radiative processes.
In this work we derive an expression which relates the shear viscosity coefficient of an ultrarelativistic gas to the total elastic and inelastic cross sections in the medium. For our derivation we consider isotropic, i.e. momentum-independent, matrix elements for the collisional processes, thus bypassing the conceptual difficulties connected with the momentum-dependent matrix elements in a (perturbative) QCD medium. To our best knowledge, analytic derivation of the shear viscosity for inelastic processes with isotropic differential cross sections is completely new.
This paper has the following structure. In Section II we start with the derivation of an expression for the shear viscosity coefficient, for which we use the entropy production in a system off equilibrium. In order to obtain a closed analytic expression for the shear viscosity coefficient in terms of the involved total cross sections, kinematic integrals will be evaluated. This is discussed in Section III, where the final expression is also presented. In Section IV the results of the analytic formula are compared with Green-Kubo based kinetic transport calculations of the shear viscosity coefficient. Finally, Section V contains summary and conclusions.
Ii Derivation of the expression for the shear viscosity coefficient
The derivation we follow is built upon the entropy principle, i.e. the second law of thermodynamics is used. In presence of dissipative fields, of which we will consider only the shear stress tensor throughout this work, the entropy production must have a non-negative form, and hence can be written as discussed e.g. in Ref. El:2010mt ():
In the latter expression denotes the shear tensor and the shear viscosity coefficient; is the temperature and the fugacity, which are both related to the chemical potential, . denotes a source for particle production and annihilation and vanishes in the situation where the net particle number does not change.
Note that in Eq. (1) is identified with the shear viscosity since one can show that in the lowest-order, i.e. Navier-Stokes, dissipative hydrodynamic theory, where , the entropy production indeed takes the form (1). Since higher-order theories (such as Israel-Stewart IS ()) are supposed to converge to the Navier-Stokes form, the entropy production is always written in the form (1) and a consistent dynamic equation for can be found, like e.g. in Ref. Muronga:2003ta ().
On the other hand, the entropy production can be obtained from the total divergence of the entropy current defined in the kinetic theory as
where denotes the phase-space distribution function and . For systems away from kinetic equilibrium the distribution can be written introducing a small deviation from the equilibrium form :
Some constrains must be applied to Eq. (3) in order to preserve consistency. For instance, if moments of from (3) are calculated, the particle number four-vector and the energy-momentum tensor must be recovered. This means that the deviation must depend on the deviations and of the energy-momentum tensor resp. particle number four-vector from their ideal forms. The explicit functional dependence is given by the following expression, as obtained in scope of Grad’s method of moments Grad49 (); DeGroot (); El:2008yy (); Muronga:2006zx ():
Note that the momentum-dependence of can be extended beyond the second power, as demonstrated in Ref. Denicol:2012cn (). However, our derivation is restricted to the original approach of Grad, for which the series must be truncated at second power of momentum. In the kinetic theory, dynamics of is governed by a transport equation, such as the Boltzmann Equation
with the functional on the right hand side denoting the collision term. Combining Eqs. (2), (3), (5) and (1) one obtains the following expression for the shear viscosity coefficient El:2008yy (); Muronga:2006zx ()
with denoting the second moment of the collision term
An obvious property of the obtained expression is the explicit dependence on the shear tensor components. But since the shear viscosity coefficient is a medium property and should not depend on the dynamics encoded in , Eq. (6) should be considered in the limit .
The functional contains information about the collisional processes in the medium. We consider only local interactions between constituents of the medium and assume them to be Boltzmann-particles, i.e. quantum statistics effects are neglected. If both elastic and inelastic interactions are considered, the second moment of the collision term takes the following form:
with . The first two terms in (LABEL:eq:collterm) account for elastic processes, whereas the last four terms capture the inelastic and reverse processes. The numerical factors and account for multiple counting of identical Boltzmann particles.
In order to evaluate the integrals in (LABEL:eq:collterm) analytically, the distribution functions will be replaced by the near-equilibrium approximations (3) and only terms up to first order in will be considered:
The equilibrium distribution is the Boltzmann distribution
with the degeneracy factor .
Analytic evaluation of the integrals in (LABEL:eq:collterm) can be simplified if a specific symmetry for the geometry of the system is considered. Since the shear viscosity coefficient is a material property, its value cannot depend on the form of the energy-momentum tensor distortion, i.e. any plausible form for the shear tensor can be considered to evaluate (LABEL:eq:collterm). The simplest geometry that can be considered for the shear tensor, which has to be traceless, is a diagonal from with one independent component :
This form of the shear tensor implies deviation from isotropy in -direction and isotropy in the transverse plane. Systems with this kind of symmetry have been studied e.g. in Refs. Bouras:2010hm (); El:2010mt (). With the shear tensor given by Eq. (11), the expression for the shear viscosity (6) as well as the collision term moment (LABEL:eq:collterm) are significantly simplified, since now is the only unknown parameter, for which the limit must be considered. In fact, with the introduced form of the shear pressure tensor the correction (4) to the distribution function becomes (for sake of compactness we will suppress the dependence of on in the notation)
Now the contraction in (6) can be rewritten as follows
The parts of the integrals that do not contain (i.e. those parts where the integration goes over and not ) have to vanish, since there is no entropy production in an equilibrated medium. The remaining integrals will then be of order , just as the expression in the numerator of (6), so that the shear viscosity coefficient will not depend on , as it should be.
Iii Evaluation of the kinematic integrals
In general the matrix elements will have a highly non-trivial dependence on momenta of the colliding particles and analytic evaluation might be difficult. Here we will consider angle-independent matrix elements to obtain the shear viscosity as function of total cross section. The total cross sections and for elastic resp. inelastic two-particle processes can be defined according to byckling73 (); Xu:2004mz ():
For these definitions the center-of-mass energy squared of the colliding particles has been introduced. For a process with particles in the initial state a classical cross section cannot be defined, but one can define an analogous quantity Xu:2004mz ()
The squared matrix elements and depend on each other by virtue of the detailed balance requirement:
If the matrix elements are angle-independent, i.e. if the scattering is isotropic, as considered for the following derivations, one can obtain simple relations between them, the cross sections , and the quantity Xu:2004mz () by performing the momentum integration in the definitions (14) – (16). Note that the right-hand-sides of Eqs. (14) – (15) are Lorentz-invariant and thus can be conveniently evaluated in the center-of-mass frame of colliding particle pair:
Here, again, is the degeneracy factor of the constituent particles.
With the definitions (14) – (16) half of the terms (the second, fifth and sixth terms) in (LABEL:eq:collterm) are significantly simplified. Calculation of the remaining terms is a bit more involved. The first, third and fourth terms in (LABEL:eq:collterm) contain integrals over the or final state momenta which can be formally written as
In the latter expression denotes a (polynomial) function of the final state momenta. For the matrix elements the relations (18) – (20) can be used. However it is important to note that the expression (21) is not anymore Lorentz-invariant due to presence of the function . At least for the terms describing a two-particle initial state it is still convenient to transform the integral into the center-of-mass frame for evaluation and to apply a back-transform afterwards. This means rewriting Eq. (21) in the following form
where denotes quantities in the center-of-mass frame, in which and . To rewrite (21) into the form (22) Lorentz invariance of and of the 4-dimensional Delta-function can be used. Notation means that function acts on momenta that are Lorentz-transformed from the center-of-mass frame into the lab frame:
with the Lorentz boosts
For the term describing a three-particle initial state (the fourth term in (LABEL:eq:collterm) resp. (LABEL:eq:momentofcollterm)) such transformation into and then back from the center-of-mass frame does not induce any significant simplification of the calculation so that the integration must be evaluated directly in the lab frame.
After evaluating all the integrals we finally obtain the following expression for the shear viscosity coefficient:
The second and the third terms in the denominator of the latter equation account for contributions of resp. processes. The factor in the term denotes the fugacity, which is a measure for the degree of saturation of the particle density in a system
with . Setting (or, accordingly, in Eq. (10)), i.e. considering a chemically equilibrated system, we obtain
Iv Comparison with kinetic transport calculations
In order to confirm the validity of Eq. (29) it is necessary to extract the transport coefficients from the full solution of the relativistic Boltzmann equation. An accurate numerical solver of the relativistic Boltzmann equation is the Boltzmann Approach to Multi-Parton Scatterings (BAMPS) Xu:2007aa (); Xu:2004mz (). This has been demonstrated in simulating shock waves in various scenarios Bouras:2009nn (); Bouras:2012mh (). The success of those approaches then motivated to compare different approaches of dissipative hydrodynamics Bouras:2010hm (); Denicol:2012vq () and relativistic lattice Boltzmann calculations oai:arXiv.org:1009.0129 (); oai:arXiv.org:1109.0640 () to the numerical solutions of BAMPS in order to verify their validity.
The accurate solution of the relativistic Boltzmann equation allows the numerical extraction of the transport coefficient like the shear viscosity and heat conductivity from BAMPS. This has been demonstrated using the Green-Kubo formalism, which has been successfully applied to extract the shear viscosity to a very high precision from BAMPS calculations Wesp:2011yy () and agreed excellently with extraction of the shear viscosity coefficient using the classical picture of a velocity gradient Reining:2011xn (). Furthermore, in a recent work the extraction of heat conductivity from BAMPS calculations could clarify which theoretical prediction of the heat conductivity coefficient each originating from different derivations of dissipative hydrodynamics is the most accurate Greif:2013bb (). In the following we apply the Green-Kubo formalism to extract the shear viscosity from BAMPS calculations in order to verify the accuracy of Eq. (29).
The Green-Kubo relation allows to connect a linear transport coefficient with the correlation function of its current. The evaluation of the shear viscosity using the Green-Kubo method is realized by calculating the correlator of the shear-stress tensor in a thermal and equilibrated without any spatial gradients.
The extraction of the shear-stress tensor from BAMPS is evaluated by calculating the energy-momentum tensor defined as
In BAMPS this is realized by summing up over all discrete particle momenta in the volume averaged simulation
Here, is the number of particles, while is the volume of the system and is the test particle number Xu:2004mz (); Bouras:2010hm (). Due to the fact that we consider a stationary system, the corresponding fluctuations of the energy-momentum tensor are of the same order as the one of the shear-stress tensor and therefore the shear-stress tensor has not to be calculated explicitly. This is realized in the fact that the correlation function of energy-momentum tensor is the same as the one from the shear-stress tensor in equilibrium:
with the correlation function:
Finally, with a thermal prefactor, the corresponding Green-Kubo relation for the shear viscosity has the following form Wesp:2011yy ():
where denotes the ensemble averaged correlation function in thermal equilibrium of the shear components at time and at . The correlator is summed over all spatial components and . In case of stochastic processes, the correlation function in Eq. (34) will have the form of an exponential. More details to the Green-Cubo method and the extraction within BAMPS is shown explicitly in Wesp:2011yy ().
In Fig. 1 we demonstrate the shear viscosity values extracted this way from BAMPS calculations for which only inelastic and processes were considered. The shear viscosity is shown as function of the inelastic cross section . We observe a very good agreement of the results extracted from BAMPS (symbols) with the results obtained using Eq. (29) (lines) derived in this work. We also would like to mention that Eq. (29) confirms the values of the shear viscosity which we obtained in Ref. El:2010mt () by numerical evaluation of the Equation (6). Equation (29) implies that if the total cross sections are equal, relaxation of the medium towards equilibrium will proceed times faster if only inelastic processes are involved as compared to the case when only elastic processes are involved (since the hydrodynamic relaxation time is directly proportional to the shear viscosity coefficient – comp. Muronga:2003ta (); El:2009vj ()). This reflects the fact that the inelastic processes are more efficient in driving the medium towards equilibrium, as was demonstrated in Refs. Xu:2007aa (); El:2008yy (); El:2010mt ().
We have derived an analytic expression relating the shear viscosity coefficient of an ultra-relativistic gas to the total elastic and inelastic cross sections of isotropic scattering processes in the medium. From the obtained expression one recognizes that, given all total cross sections are equal, isotropic inelastic processes are times more efficient than the elastic ones in restoring kinetic equilibrium in a system. The values of the shear viscosity obtained by Eq. (29) are in very good agreement with the results of Green-Kubo based calculations performed using the kinetic transport model BAMPS. The correspondence between shear viscosity and total cross sections of elastic and inelastic processes reported is another step to establish a connection between the kinetic transport and dissipative hydrodynamic models.
This work was supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse. BAMPS calculations were performed using the resources of the Center for Scientific Computing of the Goethe University Frankfurt.
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