Exact solutions of the (0+1)-dimensional kinetic equation in the relaxation time approximation
aa r X i v : . [ h e p - ph ] F e b Exact solutions of the (0+1)-dimensional kineticequation in the relaxation time approximation
Ewa Maksymiuk
Institute od Physics, Jan Kochanowski University, PL-25460 Kielce, PolandE-mail:
Abstract.
We present exact solutions of the (0+1)-dimensional kinetic equation for a massivegas in the relaxation time approximation. At first, we analyse the case of classical statisticsand argue that the traditional second-order hydrodynamics misses the shear-bulk coupling. Inthe next step, we include Bose-Einstein and Fermi-Dirac statistics in the calculations and showthat they are important for description of the effects connected with bulk viscosity.
1. Introduction
In this paper we present exact solutions of the (0+1)-dimensional kinetic (Boltzmann) equationfor a massive gas in the relaxation time approximation [1]. Our results describe a purelylongitudinal boost-invariant expansion and may be useful for description of very early stagesof relativistic heavy-ion collisions. At first, we analyse the case of classical statistics and showthat the traditional second-order hydrodynamics has problems to correctly reproduce the kineticresult. This discrepancy is due to the missing shear-bulk coupling in the standard second-orderhydrodynamics. We further take into account the effects of quantum statistics [2]. They turnout to be important for the bulk viscous pressure.The results presented here generalise several earlier results obtained for massless particles [3,4]. We note that the exact solutions of the kinetic equation help us to select the right form ofthe kinetic coefficients [3, 4] and the correct structure of the hydrodynamic equations, as hasbeen demonstrated recently in Refs. [3, 4, 5, 6, 7, 8, 9].
2. The boost-invariant Boltzmann equation in the relaxation time approximation
Our approach uses a simple form of the kinetic equation, namely p µ ∂ µ f = C [ f ] , C [ f ] = − p α u α τ eq ( f − f eq ) , (1)where f is the one-particle phase-space distribution function depending on the parton space-time coordinates x and momentum p , and C is the collision term written in the relaxation timeapproximation [10, 11, 12]. The parameter τ eq is the relaxation time. In our present calculationswe use the value τ eq = 0 .
25 fm / c. The boost-invariance implies that the kinetic equation (1)can be rewritten in the form ∂f∂τ = f eq − fτ eq , (2)here τ = √ t − z is the proper time. In addition, the function f may depend only on thethree variables: τ , w and p T , where w = tp L − zE . The background equilibrium distributionfunction may be written as f eq ( τ, w, p T ) = 2(2 π ) exp q w + (cid:0) m + p T (cid:1) τ T ( τ ) τ − ǫ − . (3)In Eq. (3) the parameter ǫ specifies the appropriate quantum statistics. With ǫ = +1 , , − T µν ( τ ) = g Z dP p µ p ν f ( τ, w, p T ) , ∂ µ T µν = 0 . (4)Here g is the number of internal degrees of freedom and dP is the momentum integrationmeasure. Equation (4) is fulfilled if the energy densities calculated with the distribution functions f and f eq are equal. This leads us to our main equation T ( τ ) ˜ H (cid:20) , mT ( τ ) (cid:21) (5)= D ( τ, τ )Λ ˜ H (cid:20) τ τ √ ξ , m Λ (cid:21) + τ Z τ dτ ′ τ eq D ( τ, τ ′ ) T ( τ ′ ) ˜ H (cid:20) τ ′ τ , mT ( τ ′ ) (cid:21) . This is an integral equation for the effective temperature T ( τ ) that can be solved using theiterative method [13]. The function ˜ H ( y, z ) is defined in Ref. [2]. Here we used the initialcondition given by the Romatsche-Strickland form f ( w, p T ) = 14 π exp q (1 + ξ ) w + ( m + p T ) τ Λ τ − ǫ − . (6)We note that the form of Eq. (5) is the same for the three different statistics. The differencesare hidden in the implicit dependence of the functions f eq , f , and ˜ H on the quantum statisticsparameter ǫ .
3. Shear and bulk viscosities of a relativistic quantum massive gas
To find the shear viscosity η for the Bose–Einstein and Fermi–Dirac gases, we use the formula [14] η ( T ) = 2 g τ eq T Z d p (2 π ) p E f eq (1 + ǫf eq ) . (7)On the other hand, we determine the effective shear viscosity using the exact solution of thekinetic equation η eff ( τ ) = 12 τ [ P T ( τ ) − P L ( τ )] . (8)We treat the bulk viscosity ζ in the similar way as the shear viscosity. For a quantum massivegas, the formula for the bulk viscosity is the following [14] ζ ( T ) = 2 g τ eq T Z d p (2 π ) m E f eq (1 + ǫf eq ) c s E − p E ! . (9) =
300 MeVm =
300 MeV Τ = (cid:144) c Τ eq = (cid:144) c Ξ = H a L - - - - Τ @ fm (cid:144) c D - Ζ @ f m - D T =
300 MeVm =
300 MeV Τ = (cid:144) c Τ eq = (cid:144) c Ξ = H b L Τ @ fm (cid:144) c D Figure 1.
Comparison of the bulk viscosity for Boltzmann statistics. The panel (a) describesa system which is initially isotropic, while the panel (b) describes a system which is initiallyhighly oblate. Red solid lines represent the effective bulk viscosity obtained from the kinetictheory, blue-dashed lines describe the bulk viscosity given by Eq. (9), finally, black lines showour results obtained for three versions of the second–order hydrodynamics [1].To find the effective bulk viscosity, which is obtained from the exact solution we use the formula ζ eff ( τ ) = − τ [ P L ( τ ) + 2 P T ( τ ) − P eq ( τ )] . (10)The sound velocity appearing in Eq. (9) is obtained from the formula c s ( T ) = ∂ P eq ( T ) /∂ E eq ( T ).In the classical limit, ǫ →
0, the integrals (7) and (9) become analytic. The appropriate formulascan be found in Ref. [1].
4. Results
Results shown in Fig. 1 represent the time evolution of the bulk viscosity calculated directly fromthe kinetic theory and compared with three formulations of the second-order hydrodynamics [1].In the case of an initially isotropic system we can see that the simplest formulation of the second-order hydrodynamics (dotted line) gives the worst agreement with the kinetic theory. On theother hand, this formulation gives the best agreement for an initially highly oblate system. Theproblems illustrated in Fig. 1 helped to identify the importance of the shear-bulk couplings[15, 16, 17] in the hydrodynamic approach. To have a good agreement it is necessary to useequations where the bulk and shear viscosities are correlated [16, 17]. The proper description ofthe bulk pressure is important as it may affect different physical observables studied in relativisticheavy-ion collisions [18, 19, 20].To show the effects of quantum statistics on the evolution of matter we calculated the shearviscosity using Eqs. (7) and (8) and the bulk viscosity using Eqs. (9) and (10). In Fig. 2 wepresent our results. In the case of the shear viscosity, we find only small differences between theresults obtained for Bose-Einstein, Boltzmann, and Fermi-Dirac statistics. This is in contrast tothe bulk-viscosity case, where we find important differences, especially, for the initially isotropicsystems.
5. Conclusions
We have constructed the exact solutions of the Boltzmann equation using analytical andnumerical methods. This allowed us to find the effective bulk and shear viscosities of the(0+1)–dimensional system and to compare them with the analytic formulas appearing in theliterature. We have shown that standard equations of the second-order hydrodynamics do notwork properly — although the quantum statistics effects are not essential for the shear viscositythey become quite important for the correct description of the bulk viscous effects. =
300 MeVm =
300 MeV Τ = (cid:144) c Τ eq = (cid:144) c Ξ = H a L B - EBF - D1.0 10.05.02.0 3.01.5 7.00.101.000.500.202.000.300.151.500.70 Η @ f m - D T =
300 MeVm =
300 MeV Τ = (cid:144) c Τ eq = (cid:144) c Ξ = H b L B - EBF - D1.0 10.05.02.0 3.01.5 7.00.10.20.51.02.05.0 H c L B - EBF - D1.0 10.05.02.0 3.01.5 7.0 - - - - - - Τ @ fm (cid:144) c D - Ζ @ f m - D H d L B - EBF - D1.0 10.05.02.0 3.01.5 7.00.000.050.100.150.200.250.300.35 Τ @ fm (cid:144) c D Figure 2.
Comparison of exact solutions for different quantum statistics. The values of theparameters are displayed in the panels. Panels (a) and (c) differ from (b) and (d) by the valueof the initial anisotropy parameter.
Acknowledgments
I would like to thank my supervisor Wojciech Florkowski for valuable advices and interestingdiscussions. This work was supported by Polish National Science Center grant No. DEC-2012/06/A/ST2/00390.
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