On the contrary, the ratio of shear viscosity to enthalpy density displays a difference when going to higher chemical potential. The difference between the two ratios highlights that the inclusion of the baryonic chemical potential term in the entropy calculation see Eq.

For future reference and to help shed some light on the various features of Fig.

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The top panel of Fig. Increasing chemical potential also increases shear viscosity at equal temperature for all temperatures, which is also expected. The bottom panel of Fig. Since both of these quantities depend primarily on the energy density of the system, it comes as no surprise that increasing the temperature or baryon chemical potential leads to large increases here as well.

This can at least partly explain the shape of the corresponding curves in Fig. In Fig. One should first note that the overall profile of these curves is relatively similar to those of Fig. This is expected, since as seen on Fig. At higher temperatures, there also appears to be a trend of slightly increasing relaxation time as the baryonic chemical potential increases.

### Reward Yourself

If we consider that baryons generally have smaller cross-sections than their mesonic counterparts, this easily explains the observed trend. As one can see, we observe for all temperatures a slightly increasing plateau at these values of chemical potential; note that within error bars, this calculation is still consistent with no increase at all.

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Notice that for the current range of temperatures and baryon chemical potentials, it has been checked that the use of Fermi-Dirac instead of Boltzmann statistics has a negligible effect on the observables. As one can readily see, the slope of the function gets steeper with rising temperature; this was directly visible from the previous Fig.

The slightly non-exponential behavior that one observes is investigated in more detail in Rosentg. In this section, let us first summarize previous calculations of the shear viscosity over entropy density ratio of a hadron gas and then discuss in detail how they compare with our results. As mentioned earlier, the shear viscosity of the hadron gas is an active subject of discussion, and multiple calculations of its value were performed previously, especially for the zero baryon chemical potential case. A comparison of available calculations is presented in Fig.

The Rougemont et al.

Ozvenchuk et al. Let us know discuss the comparisons for each result starting from the low temperature region. As seen on Fig. The Moroz calculation Morozvd employs an approach to calculate viscosity analytically from the relaxation time approximation in the full hadron gas.

The cross sections are implemented in analytical expressions for the shear viscosity obtained from a Chapman-Enksog expansion of the Boltzmann equations. Although information from resonances is encoded in the cross sections, the collision kernel only contains elastic processes, and no retardation effects from finite lifetimes are considered.

In Ozvenchuk et al. Ozvenchukkh the relaxation time approximation is used to simplify the Boltzmann equation and obtain a simple formula for the shear viscosity. Even when resonance formation is implemented in PHSD simulations, the relaxation time is identified with the mean-free time extracted from collision rates in a box simulation.

In this approach, the relaxation time contains no feedback from the resonance lifetimes. Although exact values differ quite a bit, the general consensus appears to confirm the expectation that the viscosity should generally decrease when approaching a phase transition.

That being said, two tendencies are appearing in this plot: some calculations are constantly decreasing for the available data in this range of temperatures, and others appear to saturate at some point and form a plateau at higher temperature; our calculation is among the latter. Even though our results are otherwise somewhat smaller, this similarity in the behavior is striking when compared to the other tendency, which predicts a steadier decrease to sometimes much lower values around the critical temperature.

## Electric discharge

In contrast, almost all other calculations use point-like interactions for a great portion of the considered hadronic interactions, if not all. However, on inspection, this elastic cross section turns out to be much smaller than the non-elastic resonance contribution maximum of 1. In consequence, only a small number of collisions are point-like in UrQMD. Rougemont et al. Rougemonttlu use the completely different framework of holography, and is therefore excluded from this categorization.

To understand how resonance lifetimes affect the relaxation dynamics, consider a system without physically present resonances. The relaxation time is the characteristic time in which the system approaches equilibrium after a slight departure from it. This time is of microscopic origin, and it is assumed to be of the order of the collision time or the inverse of the scattering rate. Under a shear perturbation, particles with different momentum will collide redistributing their energy to approach the thermal distribution.

This collision occurs on a time scale of the order of the mean free time, and therefore the relaxation time should be of the same order. If the lifetime of the resonances is finite, but much smaller than the mean free time, then the same picture holds, because the resonance will decay long before the next collision is expected to happen.

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What happens to this picture if resonance lifetime is comparable or larger to the mean free time? Then the transport process is blocked until the resonance eventually decays, because it is only at that instant that the momentum exchange is finally performed. This picture breaks down if a sufficient portion of the interactions are point-like. To see if the physical picture that we are depicting holds, let us also apply constant isotropic cross-sections to all interactions in SMASH, so that a significant portion of the collisions will now be point-like.

## Fresno First & CPR Trai... Lib

The first one is that all points are now at a lower value of shear viscosity, which is explained by the increase in all cross-sections. The second difference is more interesting, and concerns the profile of the curve: rather than saturating at a given value, it now decreases constantly for this range of temperatures, which is what we would expect from a system in which a large part of the interactions is now point-like, so that the relaxation time is not affected by the lifetime of particles anymore. As seen on the bottom panel of Fig.

When one includes a large number elastic point-like collisions into SMASH, the plateau disappears and we recover a linear dependency of order 1, even at high temperatures. This is once again in line with the expectations of our resonance lifetime hypothesis. As a final remark, let us now considers the case of non-zero baryon chemical potential, where literature proves to be a lot scarcer, although not inexistent. In this regard we present two comparisons with other calculations Fig. In both cases, they observe a difference between the zero and non-zero baryochemical potential results, with the non-zero case yielding a smaller viscosity.

In our calculation, both cases are constant within errors. After successfully testing the calculation of the viscosity in the recently developed SMASH transport code in the simplest scenario of a single species interacting via constant isotropic cross section, we presented a detailed analysis of the shear viscosity of a hadron gas. Nicola Manini.

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