Review PaperNo Access

NEW DEVELOPMENTS IN RELATIVISTIC VISCOUS HYDRODYNAMICS

PAUL ROMATSCHKE

Institute for Nuclear Theory, University of Washington, Box 351550, Seattle, WA, 98195, USA

https://doi.org/10.1142/S0218301310014613Cited by:390 (Source: Crossref)

Abstract

Starting with a brief introduction into the basics of relativistic fluid dynamics, I discuss our current knowledge of a relativistic theory of fluid dynamics in the presence of (mostly shear) viscosity. Derivations based on the generalized second law of thermodynamics, kinetic theory, and a complete second-order gradient expansion are reviewed. The resulting fluid dynamic equations are shown to be consistent for all these derivations, when properly accounting for the respective region of applicability, and can be applied to both weakly and strongly coupled systems. In its modern formulation, relativistic viscous hydrodynamics can directly be solved numerically. This has been useful for the problem of ultrarelativistic heavy-ion collisions, and I will review the setup and results of a hydrodynamic description of experimental data for this case.

Keywords:

PACS: 12.38.Mh, 25.75.Ld, 47.75.tf

You currently do not have access to the full text article.
Recommend the journal to your library today!

References

L. Euler, Mém. Acad. Sci. Berlin 11 (1755); also in Opera omnia 12 (1907) 54 . Google Scholar
L. D. Landau and E. M. Lifshitz , Fluid Mechanics , 2nd edn. , Course of Theoretical Physics 6 ( Elsevier , 1987 ) . Google Scholar
C. L. M. H. Navier, Mém. Acad. Sci. Inst. France 6, 389 (1822). Google Scholar
G. G. Stokes, Trans. Camb. Philos. Soc. 8, 287 (1845). Google Scholar
D. T. Son and A. O. Starinets, JHEP 0603, 052 (2006), arXiv:hep-th/0601157. ADS, Google Scholar
D. H. Rischke , arXiv:nucl-th/9809044 . Google Scholar
A. Muronga, Phys. Rev. C 69, 034903 (2004), DOI: 10.1103/PhysRevC.69.034903. Web of Science, Google Scholar
W. A. Hiscock and L. Lindblom, Phys. Rev. D 31, 725 (1985), DOI: 10.1103/PhysRevD.31.725. Web of Science, ADS, Google Scholar
P. Kostadt and M. Liu, Phys. Rev. D 62, 023003 (2000), DOI: 10.1103/PhysRevD.62.023003. Web of Science, Google Scholar
J. C. Maxwell, Phil. Trans. R. Soc. 157, 49 (1867), DOI: 10.1098/rstl.1867.0004. Google Scholar
C. Cattaneo, Atti Sem. Mat. Fis. Univ. Modena 3, 3 (1948). Google Scholar
D. Jou, J. Casas-Vazquez and G. Lebon, Rep. Prog. Phys. 51, 1105 (1988), DOI: 10.1088/0034-4885/51/8/002. Web of Science, ADS, Google Scholar
D. D. Joseph and L. Preziosi, Rev. Mod. Phys. 61, 41 (1989), DOI: 10.1103/RevModPhys.61.41. Web of Science, ADS, Google Scholar
C. C. Ackermanet al., Phys. Rev. Lett. 16, 789 (1966), DOI: 10.1103/PhysRevLett.16.789. Web of Science, ADS, Google Scholar
I. Müller, Z. Phys. 198, 329 (1967). ADS, Google Scholar
W. Israel, Ann. Phys. 100, 310 (1976). Web of Science, ADS, Google Scholar
D. Jou, J. Casas-Vazquez and G. Lebon, Rep. Prog. Phys. 62, 1035 (1999), DOI: 10.1088/0034-4885/62/7/201. Web of Science, Google Scholar
R. Loganayagam, JHEP 0805, 087 (2008), arXiv:0801.3701. ADS, Google Scholar
S. Bhattacharyyaet al., JHEP 0806, 055 (2008), arXiv:0803.2526. ADS, Google Scholar
P. Romatschke, Class. Quant. Grav. 27, 025006 (2010), DOI: 10.1088/0264-9381/27/2/025006. Web of Science, Google Scholar
L. D. Landau and E. M. Lifshitz , Physical Kinetics , Course of Theoretical Physics 10 ( Pergamon Press , 1981 ) . Google Scholar
L. Boltzmann, Sitzb. d. Akad. d. Wiss. Wien 66, 275 (1872). Google Scholar
S. R. de Groot , W. A. van Leeuwen and C. G. van Weert , Relativistic Kinetic Theory: Principles and Applications ( Elsevier , North-Holland , 1980 ) . Google Scholar
W. Israel and J. M. Stewart, Phys. Lett. A 58 (1976) 213; ibid Ann. Phys. 118 (1979) 341 . Google Scholar
S. Chapman and T. G. Cowling , The Mathematical Theory of Non-uniform Gases ( Cambridge University Press , 1970 ) . Google Scholar
H. Grad, Comm. Pure Appl. Math. 2, 331 (1949), DOI: 10.1002/cpa.3160020403. Web of Science, Google Scholar
R. Baier, P. Romatschke and U. A. Wiedemann, Phys. Rev. C 73, 064903 (2006), DOI: 10.1103/PhysRevC.73.064903. Web of Science, Google Scholar
R. Baier, P. Romatschke and U. A. Wiedemann, Nucl. Phys. A 782, 313 (2007), DOI: 10.1016/j.nuclphysa.2006.10.016. Web of Science, ADS, Google Scholar
R. Baieret al., JHEP 0804, 100 (2008), arXiv:0712.2451. ADS, Google Scholar
M. J. Duff, Class. Quant. Grav. 11, 1387 (1994), DOI: 10.1088/0264-9381/11/6/004. Web of Science, ADS, Google Scholar
O. Aharonyet al., Phys. Rept. 323, 183 (2000), DOI: 10.1016/S0370-1573(99)00083-6. Web of Science, ADS, Google Scholar
J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231; ibid. Int. J. Theor. Phys. 38 (1999) 1113 . Google Scholar
D. T. Son and A. O. Starinets, Ann. Rev. Nucl. Part. Sci. 57, 95 (2007), DOI: 10.1146/annurev.nucl.57.090506.123120. Web of Science, ADS, Google Scholar
S. Bhattacharyyaet al., JHEP 0802, 045 (2008), arXiv:0712.2456. ADS, Google Scholar
M. Natsuume and T. Okamura, Phys. Rev. D 77, 066014 (2008), DOI: 10.1103/PhysRevD.77.066014. Web of Science, Google Scholar
M. Natsuume and T. Okamura, Prog. Theor. Phys. 120, 1217 (2008), DOI: 10.1143/PTP.120.1217. Web of Science, ADS, Google Scholar
M. Van Raamsdonk, JHEP 0805, 106 (2008), arXiv:0802.3224. ADS, Google Scholar
M. Haack and A. Yarom, JHEP 0810, 063 (2008), arXiv:0806.4602. ADS, Google Scholar
S. Bhattacharyyaet al., JHEP 0812, 116 (2008), arXiv:0809.4272. ADS, Google Scholar
M. Natsuume, Phys. Rev. D 78, 066010 (2008), DOI: 10.1103/PhysRevD.78.066010. Web of Science, Google Scholar
P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005), DOI: 10.1103/PhysRevLett.94.111601. Web of Science, ADS, Google Scholar
J. Erdmengeret al., JHEP 0901, 055 (2009), arXiv:0809.2488. ADS, Google Scholar
M. Haack and A. Yarom, Nucl. Phys. B 813, 140 (2009), DOI: 10.1016/j.nuclphysb.2008.12.028. Web of Science, ADS, Google Scholar
A. Buchel, Nucl. Phys. B 803, 166 (2008), DOI: 10.1016/j.nuclphysb.2008.05.024. Web of Science, ADS, Google Scholar
A. Buchel and M. Paulos, Nucl. Phys. B 805, 59 (2008), DOI: 10.1016/j.nuclphysb.2008.07.002. Web of Science, ADS, Google Scholar
A. Buchel and M. Paulos, Nucl. Phys. B 810, 40 (2009), DOI: 10.1016/j.nuclphysb.2008.10.012. Web of Science, ADS, Google Scholar
J. P. Blaizot and E. Iancu, Phys. Rept. 359, 355 (2002), DOI: 10.1016/S0370-1573(01)00061-8. Web of Science, ADS, Google Scholar
P. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0305, 051 (2003), arXiv:hep-ph/0302165. ADS, Google Scholar
S. C. Huot, S. Jeon and G. D. Moore, Phys. Rev. Lett. 98, 172303 (2007), DOI: 10.1103/PhysRevLett.98.172303. Web of Science, ADS, Google Scholar
M. A. York and G. D. Moore, Phys. Rev. D 79, 054011 (2009), DOI: 10.1103/PhysRevD.79.054011. Web of Science, Google Scholar
P. Romatschke and D. T. Son, Phys. Rev. D 80, 065021 (2009), DOI: 10.1103/PhysRevD.80.065021. Web of Science, Google Scholar
G. D. Moore, private communication . Google Scholar
A. Buchel and R. C. Myers, JHEP 0908, 016 (2009). Google Scholar
V. M. Braun, G. P. Korchemsky and D. Mueller, Prog. Part. Nucl. Phys. 51, 311 (2003), DOI: 10.1016/S0146-6410(03)90004-4. Web of Science, ADS, Google Scholar
D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973), DOI: 10.1103/PhysRevLett.30.1343. Web of Science, ADS, Google Scholar
G. D. Moore and O. Saremi, JHEP 0809, 015 (2008). ADS, Google Scholar
P. Arnold, C. Dogan and G. D. Moore, Phys. Rev. D 74, 085021 (2006), DOI: 10.1103/PhysRevD.74.085021. Web of Science, Google Scholar
B. Betz, D. Henkel and D. H. Rischke, [nucl-th] , arXiv:0812.1440 . Google Scholar
I. Kanitscheider and K. Skenderis, JHEP 0904, 062 (2009), arXiv:0901.1487. ADS, Google Scholar
B. B. Backet al., Nucl. Phys. A 757, 28 (2005), DOI: 10.1016/j.nuclphysa.2005.03.084. Web of Science, ADS, Google Scholar
M. Chenget al., Phys. Rev. D 77, 014511 (2008), DOI: 10.1103/PhysRevD.77.014511. Web of Science, Google Scholar
E. Iancu and R. Venugopalan, , arXiv:hep-ph/0303204 . Google Scholar
M. Prakashet al., Phys. Rept. 227, 321 (1993), DOI: 10.1016/0370-1573(93)90092-R. Web of Science, ADS, Google Scholar
S. A. Bass and A. Dumitru, Phys. Rev. C 61, 064909 (2000), DOI: 10.1103/PhysRevC.61.064909. Web of Science, Google Scholar
G. Boydet al., Nucl. Phys. B 469, 419 (1996), DOI: 10.1016/0550-3213(96)00170-8. Web of Science, ADS, Google Scholar
J. D. Bjorken, Phys. Rev. D 27, 140 (1983), DOI: 10.1103/PhysRevD.27.140. Web of Science, ADS, Google Scholar
P. Danielewicz and M. Gyulassy, Phys. Rev. D 31, 53 (1985), DOI: 10.1103/PhysRevD.31.53. Web of Science, ADS, Google Scholar
H. Kounoet al., Phys. Rev. D 41, 2903 (1990), DOI: 10.1103/PhysRevD.41.2903. Web of Science, Google Scholar
A. Muronga, Phys. Rev. Lett. 88, 062302 (2002), DOI: 10.1103/PhysRevLett.88.062302. Web of Science, Google Scholar
M. P. Heller, P. Surowka, R. Loganayagam, M. Spalinski and S. E. Vazquez, [hep-th] , arXiv:0805.3774 . Google Scholar
S. Kinoshitaet al., Prog. Theor. Phys. 121, 121 (2009), DOI: 10.1143/PTP.121.121. Web of Science, ADS, Google Scholar
M. Luzum and P. Romatschke, Phys. Rev. C 78, 034915 (2008), DOI: 10.1103/PhysRevC.78.034915. Web of Science, Google Scholar
U. W. Heinz, [nucl-th] , arXiv:0901.4355 . Google Scholar
P. F. Kolbet al., Nucl. Phys. A 696, 197 (2001), DOI: 10.1016/S0375-9474(01)01114-9. Web of Science, ADS, Google Scholar
D. Kharzeev and M. Nardi, Phys. Lett. B 507, 121 (2001). Web of Science, ADS, Google Scholar
L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994), DOI: 10.1103/PhysRevD.49.2233. Web of Science, ADS, Google Scholar
L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 3352 (1994), DOI: 10.1103/PhysRevD.49.3352. Web of Science, ADS, Google Scholar
T. Lappi and R. Venugopalan, Phys. Rev. C 74, 054905 (2006), DOI: 10.1103/PhysRevC.74.054905. Web of Science, Google Scholar
P. Romatschke and R. Venugopalan, Phys. Rev. Lett. 96, 062302 (2006), DOI: 10.1103/PhysRevLett.96.062302. Web of Science, Google Scholar
K. Fukushima, F. Gelis and L. McLerran, Nucl. Phys. A 786, 107 (2007), DOI: 10.1016/j.nuclphysa.2007.01.086. Web of Science, ADS, Google Scholar
D. Kharzeev, E. Levin and M. Nardi, Nucl. Phys. A 730, 448 (2004), DOI: 10.1016/j.nuclphysa.2003.08.031. Web of Science, ADS, Google Scholar
A. Dumitru, E. Molnar and Y. Nara, Phys. Rev. C 76, 024910 (2007), DOI: 10.1103/PhysRevC.76.024910. Web of Science, Google Scholar
W. H. Press et al. , Numerical Recipes in C , 2nd edn. ( Cambridge University Press , 1992 ) . Google Scholar
R. Baier and P. Romatschke, Eur. Phys. J. C 51, 677 (2007), DOI: 10.1140/epjc/s10052-007-0308-5. Web of Science, ADS, Google Scholar
P. Romatschke, Eur. Phys. J. C 52, 203 (2007), DOI: 10.1140/epjc/s10052-007-0354-z. Web of Science, ADS, Google Scholar
P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99, 172301 (2007), DOI: 10.1103/PhysRevLett.99.172301. Web of Science, ADS, Google Scholar
C++ versions of the relativistic viscous hydrodynamic codes with and without radial symmetry, http://hep.itp.tuwien.ac.at/~paulrom/ . Google Scholar
C. M. Hung and E. V. Shuryak, Phys. Rev. C 57, 1891 (1998), DOI: 10.1103/PhysRevC.57.1891. Web of Science, Google Scholar
K. Dusling and D. Teaney, Phys. Rev. C 77, 034905 (2008), DOI: 10.1103/PhysRevC.77.034905. Web of Science, Google Scholar
P. V. Ruuskanen, Acta Phys. Polon. B 18, 551 (1987). Web of Science, Google Scholar
D. H. Rischke and M. Gyulassy, Nucl. Phys. A 608, 479 (1996), DOI: 10.1016/0375-9474(96)00259-X. Web of Science, ADS, Google Scholar
F. Karsch, K. Redlich and A. Tawfik, Eur. Phys. J. C 29, 549 (2003), DOI: 10.1140/epjc/s2003-01228-y. Web of Science, ADS, Google Scholar
C. Amsleret al., Phys. Lett. B 667, 1 (2008). Web of Science, ADS, Google Scholar
F. Cooper and G. Frye, Phys. Rev. D 10, 186 (1974), DOI: 10.1103/PhysRevD.10.186. Web of Science, ADS, Google Scholar
P. Huovinen and P. V. Ruuskanen, Ann. Rev. Nucl. Part. Sci. 56, 163 (2006), DOI: 10.1146/annurev.nucl.54.070103.181236. Web of Science, ADS, Google Scholar
P. F. Kolb, Phys. Rev. C 68, 031902 (2003), DOI: 10.1103/PhysRevC.68.031902. Web of Science, Google Scholar
J. Y. Ollitrault, Phys. Rev. D 46, 229 (1992), DOI: 10.1103/PhysRevD.46.229. Web of Science, ADS, Google Scholar
J. Adamset al., Phys. Rev. Lett. 92, 062301 (2004), DOI: 10.1103/PhysRevLett.92.062301. Web of Science, Google Scholar
M. Gyulassy and T. Matsui, Phys. Rev. D 29, 419 (1984), DOI: 10.1103/PhysRevD.29.419. Web of Science, ADS, Google Scholar
J. Sollfrank, P. Koch and U. W. Heinz, Phys. Lett. B 252, 256 (1990). Web of Science, ADS, Google Scholar
J. Sollfrank, P. Koch and U. W. Heinz, Z. Phys. C 52, 593 (1991), DOI: 10.1007/BF01562334. ADS, Google Scholar
AZHYDRO version 0.2, http://karman.physics.purdue.edu/OSCAR/ . Google Scholar
K. Adcoxet al., Nucl. Phys. A 757, 184 (2005), DOI: 10.1016/j.nuclphysa.2005.03.086. Web of Science, ADS, Google Scholar
I. Arseneet al., Nucl. Phys. A 757, 1 (2005), DOI: 10.1016/j.nuclphysa.2005.02.130. Web of Science, ADS, Google Scholar
J. Adamset al., Nucl. Phys. A 757, 102 (2005), DOI: 10.1016/j.nuclphysa.2005.03.085. Web of Science, ADS, Google Scholar
P. F. Kolb and U. W. Heinz, , arXiv:nucl-th/0305084 . Google Scholar
D. Molnar and P. Huovinen, J. Phys. G 35, 104125 (2008), DOI: 10.1088/0954-3899/35/10/104125. Web of Science, Google Scholar
H. Song and U. W. Heinz, Phys. Rev. C 78, 024902 (2008), DOI: 10.1103/PhysRevC.78.024902. Web of Science, Google Scholar
D. Teaney, Phys. Rev. C 68, 034913 (2003), DOI: 10.1103/PhysRevC.68.034913. Web of Science, Google Scholar
F. Karsch, D. Kharzeev and K. Tuchin, Phys. Lett. B 663, 217 (2008). Web of Science, ADS, Google Scholar
S. Caron-Huot, Phys. Rev. D 79, 125009 (2009), DOI: 10.1103/PhysRevD.79.125009. Web of Science, ADS, Google Scholar
H. Song and U. W. Heinz, J. Phys. G 36, 064033 (2009), DOI: 10.1088/0954-3899/36/6/064033. Web of Science, Google Scholar
M. Asakawaet al., Phys. Rev. Lett. 101, 122302 (2008), DOI: 10.1103/PhysRevLett.101.122302. Web of Science, ADS, Google Scholar
D. Teaney, J. Lauret and E. V. Shuryak, Phys. Rev. Lett. 86, 4783 (2001), DOI: 10.1103/PhysRevLett.86.4783. Web of Science, ADS, Google Scholar
T. Hiranoet al., Phys. Lett. B 636, 299 (2006). Web of Science, ADS, Google Scholar
H. Petersenet al., Phys. Rev. C 78, 044901 (2008), DOI: 10.1103/PhysRevC.78.044901. Web of Science, Google Scholar
L. Lindblom and B. J. Owen, Phys. Rev. D 65, 063006 (2002), DOI: 10.1103/PhysRevD.65.063006. Web of Science, Google Scholar
W. Zimdahl, Phys. Rev. D 53, 5483 (1996), DOI: 10.1103/PhysRevD.53.5483. Web of Science, ADS, Google Scholar
S. S. Adleret al., Phys. Rev. C 69, 034909 (2004), DOI: 10.1103/PhysRevC.69.034909. Web of Science, Google Scholar
B. Alveret al., Int. J. Mod. Phys. E 16, 3331 (2008), DOI: 10.1142/S0218301307009300. ADS, Google Scholar
STAR Collaboration (B. I. Abelev et al.), [nucl-ex] , arXiv:0801.3466 . Google Scholar
P. M. Morse and H. Feshbach , Methods of Theoretical Physics ( McGraw-Hill , 1953 ) . Google Scholar
P. M. Morse and H. Feshbach , Methods of Theoretical Physics ( McGraw-Hill , 1953 ) . Google Scholar