aa r X i v : . [ nu c l - t h ] F e b On nonadditive anisotropic relativistichydrodynamics
A.V. Leonidov
P.N. Lebedev Physical Institute, Moscow, Russia
Abstract
Non-additive generalisation of relativistic anisotropic anisotropichydrodynamics is described. In the particular case of 0+1 boost-invariant hydrodynamics additional entropy production due to non-additivity is calculated.
Developing a consistent physical picture of ultrarelativistic heavy ion col-lisions still remains an outstanding challenge, see e.g. a recent concise sum-mary of some of its relevant aspects in [1]. The focus of the present letter ison constructing a theoretical description of anisotropic strongly interactingmatter created in such collisions.A fundamental origin of strong momentum anisotropy of matter createdat early and intermediate stages of high energy heavy ion collisions is be-lieved to be in the strong momentum anisotropy of glasma [2], the densegluon system created at its early stages. A direct way of accounting forthis momentum anisotropy is in turning to anisotropic hydrodynamics, seee.g. a recent review [3]. Below we will closely follow an approach of con-structing such anisotropic hydrodynamics from kinetic theory discussed in[4, 5] . Momentum anisotropy results in new interesting effects such as, e.g.,modification of the Mach cone [7].A more subtle feature is the fact that the matter under discussion is denseand strongly interacting. On fundamental grounds one expects, in particu-lar, that that energy and entropy characterising such strongly interacting For a detailed discussion see [6]. .As our aim is in developing an approach taking into account both mo-mentum anisotropy and non-additivity, to construct non-additive anisotropichydrodynamics we need to consider a non-additive Boltzmann kinetic equa-tion for a distribution function characterised by momentum anisotropy. Inthe relaxation time approximation considered below it reads p µ ∂ µ [ f ( x, p ) q ] = − p µ u µ τ eq (cid:2) f q ( x, p, ξ | Λ) − f q eq ( x, p | Λ eq ) (cid:3) (1)where q is a parameter of the Tsallis distribution controlling the degree ofnon-additivity (see below equation(4)), τ eq is a relaxation scale and theanisotropic distribution f ( x, p, ξ | Λ) is assumed to have the Romatschke-Strikland [18] form f ( x, p, ξ | Λ) = f iso (cid:18) p + ξp z Λ (cid:19) (2)where, in turn, ξ ( t ) and Λ( t ) are time-dependent parameters determiningthe degree of momentum anisotropy and the magnitude of the momentumscale correspondingly and we have assumed that the distribution function isanisotropic only in longitudinal direction. The function f eq ( x, p, | Λ eq ) = f iso (cid:0) p / Λ (cid:1) (3)corresponds to an equilibrated isotropic state characterised by an effectivescale Λ eq . For the non-additive kinetic formalism under consideration theequilibrium distribution f eq ( x, p | Λ eq ) is of a Tsallis form f eq ( x, p ) = (cid:2) − (1 − q ) p / Λ (cid:3) / (1 − q ) , (4)where in the limit q → An alternative construction was developed in [16, 17]. N µ , energy-momentum ten-sor T µν and entropy current S µ : N µ = Z d p (2 π ) p p µ f ( x, p ) q (5) T µν = Z d p (2 π ) p p µ p ν f ( x, p ) q (6) S µ = − Z d p (2 π ) p p µ [ f ( x, p ) q ln q f ( x, p ) − f ( x, p )] (7)where ln q ( x ) = x − q − − q (8)Let us note that in the literature [13, 14, 15] one can find different suggestionsfor the generalised entropy current. The expression in (7) follows the choicemade in [14, 15].To ensure the energy-momentum conservation in (1) we employ the Lan-dau matching condition, see e.g. [3, 14, 15] for the energy ǫ = T ǫ ( ξ, Λ) = ǫ eq (Λ eq ) , (9)where calculations in the left- and right-hand side of (9) are performed withthe distribution functions (2) and (3) correspondingly leading to the followingmatching condition on Λ eq and Λ:Λ eq = R ( ξ ) / Λ , R ( ξ ) = 12 (cid:18)
11 + ξ + arctan √ ξ √ ξ (cid:19) (10)In this letter, following [14, 15], we consider the boost-invariant 0 + 1 -dimensional hydrodynamics in which all quantities depend only on the propertime τ defined by Milne coordinates ( τ, η ) defined by t = τ cosh η, z = τ sinh η (11)Hydrodynamic equations are those for the first two moments of the Boltz-mann equation for particle and energy-momentum currents (5,6). It is easyto see that the correspoding calculations for the non-additive case closely3ollow those described in [4] and result in the same evolution equations forΛ and ξ : ∂ τ ξ = 2(1 + ξ ) τ − ξ ) τ eq R ( ξ ) G ( ξ ) ∂ τ Λ = 1 + ξτ eq R ′ ( ξ ) G ( ξ )Λ (12)where G ( ξ ) = R / ( ξ ) √ ξ − R ( ξ ) + 3(1 + ξ ) R ′ ( ξ ) (13)Let us stress, that the number density (5) and energy-momentum tensor (6)in the non-additive case do of course differ from their additive counterparts.Let us turn to the analysis of the evolution of the entropy density S ≡ S . In performing the calculation it is convenient to explicitly separate thedependence on the anisotropy parameter S ( ξ, Λ) = 1 √ ξ S iso (Λ) (14)where S iso (Λ) = Λ (2 π ) Z dw w / [ f q iso ( w ) ln q f iso ( w ) − f iso ( w )] (15)and w = p / Λ . Let us write the equation for ∂ τ S in the following form: ∂ τ S = ∆ ( τ ) + ( q − q ( τ ) (16)where we have separated the additive ∆ ( τ ) and non-additive ( q − ( τ )such that in the additive limit q → ( τ ) = − (cid:20)
12 11 + ξ ( ∂ τ ξ ) −
3Λ ( ∂ τ Λ) (cid:21) S ∆ q ( τ ) = 1 √ ξ ∂ τ Λ Z dw w / Ln q f iso ( w ) (17)where Ln q f iso ( w ) ≡ q f q iso ( w ) − q − ( τ ) = 1 τ eq h R / ( ξ ) p ξ − i S (19)∆ q ( τ ) = p ξ τ eq R ′ ( ξ ) G ( ξ ) Z dw w / Ln q f iso ( w ) (20)Equations (19,20) present the main result of the paper: non-additivity indescribing collective properties of the anisotropic hydrodynamics results inadditional contribution to entropy production described by (20) on top ofthe previously known [4] contribution from momentum anisotropy describedby the equation (19).The work was supported by the RFBR project 18-02-40131. References [1] F. Gelis, [arXiv:2102.07604 [hep-ph]].[2] T. Lappi and L. McLerran, Nucl. Phys. A (2006), 200-212doi:10.1016/j.nuclphysa.2006.04.001 [arXiv:hep-ph/0602189 [hep-ph]].[3] M. Alqahtani, M. Nopoush and M. Strickland, Prog. Part. Nucl. Phys. (2018), 204-248 doi:10.1016/j.ppnp.2018.05.004 [arXiv:1712.03282[nucl-th]].[4] M. Martinez and M. Strickland, Nucl. Phys. A (2010), 183-197doi:10.1016/j.nuclphysa.2010.08.011 [arXiv:1007.0889 [nucl-th]].[5] M. Martinez and M. Strickland, Phys. Rev. C (2010), 024906doi:10.1103/PhysRevC.81.024906 [arXiv:0909.0264 [hep-ph]].[6] E. Molnar, H. Niemi and D. H. Rischke, Phys. Rev. D (2016) no.11,114025 doi:10.1103/PhysRevD.93.114025 [arXiv:1602.00573 [nucl-th]].[7] M. Kirakosyan, A. Kovalenko and A. Leonidov, Eur. Phys. J. C (2019) no.5, 434 doi:10.1140/epjc/s10052-019-6919-9 [arXiv:1810.06122[hep-ph]].[8] C. Tsallis, Introduction to nonextensive statistical mechanics: approach-ing a complex world (2009), Springer Science & Business Media.59] J. Cleymans, G. I. Lykasov, A. S. Parvan, A. S. Sorin,O. V. Teryaev and D. Worku, Phys. Lett. B (2013), 351-354doi:10.1016/j.physletb.2013.05.029 [arXiv:1302.1970 [hep-ph]].[10] J. Cleymans, M. D. Azmi, A. S. Parvan and O. V. Teryaev, EPJ WebConf. (2017), 11004 doi:10.1051/epjconf/201713711004[11] K. Shen, G. G. Barnaf¨oldi and T. S. Bir´o, Universe (2019) no.5, 122doi:10.3390/universe5050122 [arXiv:1905.08402 [hep-ph]].[12] G. B´ır´o, G. G. Barnaf¨oldi and T. S. Bir´o, J. Phys. G (2020) no.10,105002 doi:10.1088/1361-6471/ab8dcb [arXiv:2003.03278 [hep-ph]].[13] A. Lavagno, Phys. Lett. A (2002), 13-18 doi:10.1016/S0375-9601(02)00964-7 [arXiv:cond-mat/0207353 [cond-mat.stat-mech]].[14] T. S. Biro and E. Molnar, Phys. Rev. C (2012), 024905doi:10.1103/PhysRevC.85.024905 [arXiv:1109.2482 [nucl-th]].[15] T. S. Bir´o and E. Moln´ar, Eur. Phys. J. A (2012), 172doi:10.1140/epja/i2012-12172-8 [arXiv:1205.6079 [nucl-th]].[16] T. Osada and G. Wilk, Phys. Rev. C (2008), 044903 [erratum:Phys. Rev. C (2008), 069903] doi:10.1103/PhysRevC.77.044903[arXiv:0710.1905 [nucl-th]].[17] T. Osada and G. Wilk, Indian J. Phys. (2011), 941-946doi:10.1007/s12648-011-0103-x [arXiv:0805.2253 [nucl-th]].[18] P. Romatschke and M. Strickland, Phys. Rev. D68