Causal viscous hydrodynamics -- the different 2nd order formalisms

Israel-Stewart (I-S) Formalism[1,2,3]:

(used by codes from Song&Heinz (VISH2+1,OSU)[4,5], Romatsche&Romatschke (INT)[6], Huovinen&Molnar(Purdue)[7], Chaudhuri(Calcutta)[8])

* Simplified I-S formalism: (used by code from Song&Heinz[4] (VISH2+1,OSU))

• (1) $\Delta _{j}^{m}\Delta _{k}^{n}{\dot {\pi }}^{jk}=-{\frac {1}{\tau _{\pi }}}(\pi ^{mn}{-}2\eta \sigma ^{mn})$ (--simplified I-S eqn )
  

* Full I-S formalism from conformal symmetry constraint: (used by code from Romatsche&Romatschke[6](INT))

• (2) $\Delta _{j}^{m}\Delta _{k}^{n}{\dot {\pi }}^{jk}=-{\frac {1}{\tau _{\pi }}}(\pi ^{mn}{-}2\eta \sigma ^{mn})+{\frac {1}{2}}\pi ^{mn}\left(5D(\ln T)-\partial _{k}u^{k}\right)$ (--full I-S eqn-R&R)
  

* Full I-S formalism from 2nd law of thermodyamics: (used by code from Song&Heinz[5](VISH2+1,OSU))

• (3) $\Delta _{j}^{m}\Delta _{k}^{n}{\dot {\pi }}^{jk}=-{\frac {1}{\tau _{\pi }}}(\pi ^{mn}{-}2\eta \sigma ^{mn})-{\frac {1}{2}}\pi ^{mn}{\frac {\eta T}{\tau _{\pi }}}d_{k}\left({\frac {\tau _{\pi }}{\eta T}}u^{k}\right)$(--full I-S eqn-S&H)
  

Ottinger-Grmela (O-G) Formalism[9]:

(used by code from Dusling&Teaney(SUNY)[10])

In this pdf document we compare the equations for the OG and IS formalisms.

In summary the OG formalism yields:

  (1)  $\pi _{\mu \nu }=-\eta \sigma _{\mu \nu }-\tau _{2}\left[_{\langle }D\pi _{\mu \nu \rangle }+\left[{\frac {4}{3}}+{\frac {2}{3}}-{\frac {2\beta }{\alpha }}\right]\pi _{\mu \nu }\left(\nabla \cdot u\right)\right]+\tau _{2}\pi _{\lambda (\mu }\omega _{\nu )}^{\lambda }+{\frac {\tau _{2}}{\eta }}\pi _{\lambda \langle \mu }\pi _{\mu \rangle }^{\lambda }+O(\omega ^{3})$

For a conformal fluid one should choose $\beta =\alpha /3$

to do list:

• comparison between simplified I-S eqn and full I-S eqn -- Song&Heinz,arXiv:0805.1756 (nucl-th), PRC in press. 02-05/2008.
• comparison between full I-S eqn (R&R) and full I-S eqn (S&H)--from VISH2+1 (Song&Heinz), preliminary results could be found at Code verification for viscous hydrodynamics, 07/2008.
• comparison between I-S formalism and O-G formalism ---Dusling&Teaney&? in the furture.

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References:
[1] W. Israel, Ann. Phys. (NY) 100 (1976) 310; W. Israel and J. M. Stewart, Phys. Lett. 58A, 213 (1976); W. Israel and J. M. Stewart, Ann. Phys. (NY) 118 (1979) 341.
[2] A. Muronga and D. H. Rischke, arXiv:nucl-th/0407114.A. Muronga Phys. Rev. C 69, 034903 (2004); A. Muronga, Phys. Rev. C76 014909and 014910 (2007).
[3] U. W. Heinz, H. Song, A. K. Chaudhuri, Phys.Rev.C73:034904,2006.
[4] H. Song and U. Heinz, Phys.Lett.B 658, 279,(2008); Phys.Rev.C77, 064901, (2008).
[5] H. Song and U. Heinz, arXiv:0805.1756 (nucl-th), PRC in press.
[6] P. Romatschke and U. Romatschke, Phys.Rev.Lett.99, 172301 (2007). M. Luzum and P. Romatschke, arXiv:0804.4015 (nucl-th).
[7] D. Molnar, P. Huovinen, arXiv:0806.1367 (nucl-th).
[8] A. K. Chaudhuri,arXiv:0708.1752 (nucl-th); arXiv:0801.3180 (nucl-th) and arXiv:0803.0643 (nucl-th).
[9] M. Grmela, H. C. Ottinger, Phys. Rev. E 56, 6620 (1997); H. C. Ottinger, M. Grmela, Phys. Rev. E 56, 6633 (1997); H. C. Ottinger, Phys. Rev. E 57, 1416 (1998).
[10] K. Dusling, D. Teaney, Phys.Rev.C77:034905,2008.

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