View source for Causal viscous hydrodynamics -- the different 2nd order formalisms

== '''Israel-Stewart (I-S) Formalism[1,2,3]:''' ==
<BR>
(used by codes from '''Song&Heinz''' (VISH2+1,OSU)[4,5], '''Romatsche&Romatschke''' (INT)[6], '''Huovinen&Molnar'''(Purdue)[7], '''Chaudhuri'''(Calcutta)[8])<BR> 


'''*  Simplified I-S formalism:''' (used by code from '''Song&Heinz[4]''' (VISH2+1,OSU))<BR>
*   (1)  <math>\Delta^m_j\Delta^n_k\dot{\pi}^{jk}=-\frac{1}{\tau_{\pi}}(\pi^{mn}{-}2\eta\sigma^{mn})
</math> (--simplified I-S eqn )<BR>
'''*  Full I-S formalism from conformal symmetry constraint:''' (used by code from '''Romatsche&Romatschke[6]'''(INT))<BR>
*   (2)  <math>\Delta^m_j\Delta^n_k\dot{\pi}^{jk}=-\frac{1}{\tau_{\pi}}(\pi^{mn}{-}2\eta\sigma^{mn})+\frac{1}{2}\pi^{mn}\left(5 D(\ln T)-\partial_k u^k\right)
</math> (--full I-S eqn-R&R) <BR> 
'''*  Full I-S formalism from 2nd law of thermodyamics:''' (used by code from '''Song&Heinz[5]'''(VISH2+1,OSU))<BR> 
*   (3)  <math>\Delta^m_j\Delta^n_k\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)</math>(--full I-S eqn-S&H)<BR>




== '''Ottinger-Grmela (O-G) Formalism[9]:''' ==
<BR>
(used by code from '''Dusling&Teaney'''(SUNY)[10])<BR>

In [http://sites.google.com/site/kdusling/physics/viscous-hydrodynamics/vhydro2.pdf this pdf document] we compare the equations for the OG and IS formalisms.

In summary the OG formalism yields:

   (1)  <math>
\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)               
</math>

For a conformal fluid one should choose <math>\beta=\alpha/3</math>

== '''to do list:''' ==

*   comparison between '''simplified I-S eqn''' and '''full I-S eqn''' -- Song&Heinz,[http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0805.1756 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|Code verification for viscous hydrodynamics]], 07/2008.
*   comparison between '''I-S formalism''' and '''O-G formalism'''  ---Dusling&Teaney&? in the furture. 


----

'''References:''' <BR>
[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.<BR>
[2] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=nucl-th/0407114  A. Muronga and D. H. Rischke, arXiv:nucl-th/0407114.][http://www.slac.stanford.edu/spires/find/hep/www?eprint=nucl-th/0309055  A. Muronga  Phys. Rev. C 69, 034903 (2004);] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=nucl-th/0611090 A. Muronga, Phys. Rev. C76 014909]and [http://www.slac.stanford.edu/spires/find/hep/www?eprint=nucl-th/0611091 014910 (2007).]<BR>
[3] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=nucl-th/0510014 U. W. Heinz, H. Song, A. K. Chaudhuri, Phys.Rev.C73:034904,2006.] <BR>
[4] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0709.0742 H. Song and U. Heinz, Phys.Lett.B 658, 279,(2008);] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0712.3715 Phys.Rev.C77, 064901, (2008).]<BR>
[5] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0805.1756 H. Song and U. Heinz, arXiv:0805.1756 (nucl-th), PRC in press.]<BR>
[6] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0706.1522 P. Romatschke and U. Romatschke,  Phys.Rev.Lett.99, 172301 (2007).] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0804.4015 M. Luzum and  P. Romatschke, arXiv:0804.4015 (nucl-th).]    <BR>
[7] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0806.1367 D. Molnar, P. Huovinen, arXiv:0806.1367 (nucl-th).]<Br>
[8] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0708.1252 A. K. Chaudhuri,arXiv:0708.1752 (nucl-th)]; [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0801.3180 arXiv:0801.3180 (nucl-th)] and [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0803.0643 arXiv:0803.0643 (nucl-th).] <Br>
[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).<BR>
[10] [http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0710.5932 K. Dusling, D. Teaney, Phys.Rev.C77:034905,2008.]



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