New formulation of leading order anisotropic hydrodynamics
NNew formulation of leading order anisotropichydrodynamics
LeonardoTinti
Institute of Physics, Jan Kochanowski University, PL-25406 Kielce, PolandE-mail: [email protected]
Abstract.
Anisotropic hydrodynamics is a reorganization of the relativistic hydrodynamics expansion,with the leading order already containing substantial momentum-space anisotropies. Thelatter are a cause of concern in the traditional viscous hydrodynamics, since large momentumanisotropies generated in ultrarelativistic heavy-ion collisions are not consistent with thehypothesis of small deviations from an isotropic background, i.e., from the local equilibriumdistribution.We discuss the leading order of the expansion, presenting a new formulation for the (1+1)–dimensional case, namely, for the longitudinally boost invariant and cylindrically symmetricflow. This new approach is consistent with the well established framework of Israel and Stewartin the close to equilibrium limit (where we expect viscous hydrodynamics to work well). Ifwe consider the (0+1)–dimensional case, that is, transversally homogeneous and longitudinallyboost invariant flow, the new form of anisotropic hydrodynamics leads to better agreement withknown solutions of the Boltzmann equation than the previous formulations, especially when weconsider finite mass particles.
1. Introduction
Relativistic hydrodynamics plays a fundamental role in modeling of relativistic heavy-ioncollisions, see for instance Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Early calculationswere based on perfect fluid hydrodynamics, however, nowadays viscous codes are preferred. Bothbecause they provide a better description of the data and because of general arguments thatthe fluid viscosity cannot be zero, which follows from quantum mechanical considerations [15]as well as from the AdS/CFT correspondence [16]. Despite its obvious success, there are stillfundamental issues with the ordinary viscous hydrodynamics expansion. A new approach totreat these problems is anisotropic hydrodynamics (aHydro) [17, 18, 19, 20, 21, 22, 23, 24, 25,26, 27, 28, 29], where the large momentum anisotropy, providing large pressure corrections, istreated in a non perturbative way starting from the leading order of the hydrodynamics expasion.
2. The hydrodynamics expansion
The most common assumption for deriving hydrodynamics from relativistic kinetic theory is thatthe particle distribution function f ( x, p ) is very close to local equilibrium. Ignoring conservedcharges and in the Boltzmann limit we have f ( x, p ) = f eq . ( x, p ) + δf ( x, p ) , f eq . ( x, p ) = k exp (cid:20) − p · U ( x ) T ( x ) (cid:21) , (1) a r X i v : . [ h e p - ph ] N ov .010.020.050.100.200.501.002.00 E (cid:144) E ΠΗ (cid:144) S = P L (cid:144) E ΠΗ (cid:144) S = Ξ = T =
300 MeV Τ = (cid:144) c P T (cid:144) E ΠΗ (cid:144) S = ΠΗ (cid:144) S = ΠΗ (cid:144) S = ΠΗ (cid:144) S = Τ @ fm (cid:144) c D (cid:144) D ΠΗ (cid:144) S = Τ @ fm (cid:144) c D ΠΗ (cid:144) S = Τ @ fm (cid:144) c D ΠΗ (cid:144) S = Figure 1. (Color online) Comparison of viscous hydro with anisotropic hydrodynamics andsecond-order viscous hydrodynamics (figure taken from [31]).with T and U µ being the effective temperature and the fluid four velocity, respectively. Theleading order in (1), f eq . , describes the perfect fluid. The viscous correction depends only on δf which is treated as a small perturbation. However, when we consider an (almost) boost invariantflow like the one we expect in the early stages of heavy ions collisions, we encounter fundamentalproblems. The four velocity gradients are inversely proportional to the proper time, therefore,the pressure corrections become close to the equilibrium pressure, questioning the validity of theperturbative treatment.The main feature of anisotropic hydrodynamics is to treat the large momentum anisotropyin a non perturbative way starting from the leading order, namely, we write f ( x, p ) = f aniso . ( x, p ) + δ ˜ f ( x, p ) . (2)In this way, the deviation δ ˜ f from the (non isotropic and dissipative) background f aniso . canbe small enough to justify a perturbative treatment. The first formulation of aHydro used thepoint dependent version of the Romatschke-Strickland form (presented in [30]) for the leadingorder of the anisotropic expansion, which in the local rest frame (LRF) reads f aniso . ( x, p ) = k exp (cid:20) − x ) (cid:114) p T + ζ ( x ) p L (cid:21) . (3)Here Λ is the momentum scale (the effective temperature T is defined using the Landau matchingand is different from Λ in general), p T and p L are the transverse and longitudinal momenta, and ζ is the anisotropy parameter. In order to close the system of equations for the leading orderof the anisotropic expansion, one has used the four momentum conservation (the first momentof the Boltzmann equation and the Landau matching) and the particle creation equation (thezeroth moment of the Boltzmann equation). In addition, the collisional kernel was treated inthe relaxation time approximation.or a longitudinally boost invariant and transversely homogeneous system there is an exactsolution of the relativistic Boltzmann equation [31]. We show in Fig. 1 one of the plotsin [31]. The comparison is done between the exact solution (BE), Israel-Stewart theory, thenew formulation of second-order viscous hydrodynamics presented in [14], and anisotropichydrodynamics (AH). Anisotropic hydrodynamics is always very close to the exact solution,while IS is providing unphysical vanishing longitudinal pressure P L , and significant deviationsfrom the exact evolution of the temperature T and the transverse pressure P T . In the mostextreme case, even DNMR approach is not reliable.
3. New formulation of the leading order Τ (cid:80) Η (cid:64) f m (cid:45) (cid:68) Ξ (cid:61) T (cid:61)
600 MeVM (cid:61)
300 MeV Τ eq (cid:61) (cid:144) c0.5 1 2 3 4 5 7 1025102050100200 Τ (cid:64) fm (cid:144) c (cid:68) Τ (cid:80) Η (cid:64) f m (cid:45) (cid:68) Ξ (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) Τ (cid:80) Ζ (cid:64) f m (cid:45) (cid:68) Ξ (cid:61) T (cid:61)
600 MeVM (cid:61)
300 MeV Τ eq (cid:61) (cid:144) c0.5 1 2 3 4 5 7 10 (cid:45) Τ (cid:64) fm (cid:144) c (cid:68) Τ (cid:80) Ζ (cid:64) f m (cid:45) (cid:68) Ξ (cid:61) Figure 2. (Color online) Time dependence of shear and bulk viscous pressure multiplied by τ (figure taken from [34]).The anisotropic background (3) takes into account differences between the longitudinalpressure P L and the transverse pressure P T , only. However if there is a non-vanishing radial flowwe expect anisotropies even in the transverse plane. As the system evolves toward equilibrium,these corrections become more important. One way to handle non trivial transverse dynamics isto treat δ ˜ f in the anisotropic expansion (2) in a perturbative way [32]. Alternatively, we proposehere to include more dynamic effects connected with anisotropy in the leading order itself.In Ref. [33] we extended the formalism of anisotropic hydrodynamics to the (1+1)–dimensional case. We started from a generalization of the Romatschke-Strickland form, whichin the local rest frame reads f aniso . ( x, p ) = k exp (cid:20) − λ ( x ) (cid:114) (1 + ξ X ) p X + (1 + ξ y ) p Y + (1 + ξ Z ) p Z (cid:21) , (4)where Z is the longitudinal direction, and X is the direction of the transverse flow. We usedthe second moment of the Boltzmann equation, in addition to the energy and momentumconservation, in order to obtain a closed set of equations. We proved that these equationsreduce to the Israel-Stewart equations in the close to equilibrium limit, where we know thatsecond-order viscous hydrodynamics is justified.We later compared this new set of equations with the solution of the Boltzmann equationand the original prescription for anisotropic hydrodynamics [34]. There is a large improvementof the agreement with the exact solution, especially for massive particles. In Fig. 2 we show thecomparison between the new formulation (eaHydro) and the original one (saHydro). The shearevolution τ Π η is very well reproduced, while the bulk evolution τ Π ζ still shows some deviationsrom the exact solution. Note that τ is the (longitudinal) proper time, P eq . is the equilibriumpressure, Π η = (cid:16) P T − P L (cid:17) , and Π ζ = (2 P T + P L − P eq . ).
4. Conclusions
Anisotropic hydrodynamics is a reorganization of the hydrodynamic expansion around a non-isotropic background. The leading order already provides large longitudinal pressure corrections,justifying the perturbative treatment of the next to leading order in heavy ion collisions. Theoriginal prescription for the leading order of anisotropic hydrodynamics does not take intoaccount pressure anisotropies in the transverse plane, therefore requiring a next to leading ordertreatment in presence of transverse expansion. We extended the original treatment allowing forcylindrically symmetric expansion already in the leading order. The agreement with the exactsolution in the case of vanishing transverse flow has been largely improved. Bulk dynamics isnot well reproduced, however, an interesting proposal is to introduce an extra degree of freedomtaking into account the isotropic pressure corrections, see Ref. [35].
Acknowledgements
This work has been supported by Polish National Science Center grant No. DEC-2012/06/A/ST2/00390.
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