Add delta f corrections for bulk viscosity

[NGoetz]

Add bulk corrections similiar to the one used by the MUSIC particle sampler.

[yukarpenko]

What is the reference for the particular functional form of the bulk (bulk+shear) viscous corrections?

[NGoetz]

What is the reference for the particular functional form of the bulk (bulk+shear) viscous corrections?

I took inspiration from here: https://webhome.phy.duke.edu/~jp401/old_music_manual/particlization.html

[yukarpenko]

I checked the link ,and since it doesn't provide a paper reference I found it. So the expression for the bulk correction is written down as Eq. 7 in 1509.06738, and one should use the formula for tau_Pi from 10.1103/PhysRevC.90.024912 (1403.0962), Eq. 4. So I combined the two, and I don't get the same expression as you do, even assuming (1/3-c_s^2)\approx 0.184. Furthermore, units in your expression don't check out - the result is not dimensionless, whereas WviscFactor has to be dimensionless. I attach a handwritten note with the derivation. bulk_corrections_note.pdf

[yukarpenko]

Another thing: there's a hardcoded assumption that (1/3-c_s^2) ~ 0.184. It depends on the EoS and on the particlization temperature, so it shouldn't really be hardcoded. I suggest making it an extra input parameter. There is another hardcoded ratio there - e+p is assumed to be 1.15*e, which means p=0.15e. This also depends on the EoS and on particlization tempeature/energy density, but since the pressure itself is "much smaller" than the energy density in this region, the sum (e+p) doesn't change that much. This one can also make an extra input parameter, in principle - why to leave hard-coded ratios in the code?

[NGoetz]

I took a look at it again - it's been quite some time. I definitely missed a factor here. What I did was starting from 1509.06738 and using the expression for tau_Pi from vHLLE _tauPi = 6.0 / 5.068 * zetaS(e,T) / T;; - I missed putting the 1/5.068. But if I have not missed anything, that should be the only difference. I am all up for removing hard coded ratios. But I think that should be also consistent with vHLLE. Are there any assumptions in vHLLE (like for the tau_Pi) one needs to be aware of here?

[yukarpenko]

You've missed the entropy density. Note that zetaS is not zeta (bulk viscosity) but the ratio zeta/s = bulk viscosity / entropy density. And yes, there's also 1/hbarC missing. Also it is good that you brought the tauPi parametrization in vhlle up - this should be updated. Back in the days I didn't really care about bulk viscosity so I just put a factor 6 to account for some average speed of sound and what not, but it would be better to have a more proper expression there, like from 1403.0962.

[yukarpenko]

Also, in order to add the missing entropy density one has to compute it somehow. In vHLLE, EoS typically does it, but in the hadron sampler there is no EoS module attached. But, it isn't really necessary since I believe the expressions in 1509.06738 and 1403.0962 I've mentioned are more generic. There is (e+p) which reduces to Ts [product of temperature and entropy density] when mu_b=0, but when mu_b!=0, there's an extra mu_bn_b. So it would be better to use the expressions from 1509.06738 and 1403.0962 directly, as I sketched in my note.

[NGoetz]

Ah yes, you are completely right, I forgot about the entropy density. I would suggest that I update the expression here according to your notes and look up what are appropriate parameters for the c_s**2 with the EOS and the particlization temperature we typically use in the hybrid. I think it makes sense then to also update the expressions on the vHLLE side as well.

[yukarpenko]

Yes it does make sense to update the expressions for taupi and tauPi in vHLLE as well. We have added up-to-date parametrizations of zeta/s itself, but the relaxation times I missed to take care of. I'll try to do it soon.

[NGoetz]

I have added two new parameters with reasonable default values and changed the bulk correction accordingly.

[yukarpenko]

Good. Shouldn't there be a square of (1/3-c_s^2) in the part ...15(1.0/3.0-params::cs2)surf[iel].T ... (line 310)? Please double-check the expressions with respect to the reference publications.

[NGoetz]

Good catch! I double checked all expressions with the literature and it should be fine by now.