Holography of the QGP Reynolds number

Abstract

The viscosity of the Quark-Gluon Plasma (QGP) is usually described holographically by the entropy-normalized dynamic viscosity η/s. However, other measures of viscosity, such as the kinematic viscosity ν and the Reynolds number Re, are often useful, and they too should be investigated from a holographic point of view. We show that a simple model of this kind puts an upper bound on Re for nearly central collisions at a given temperature; this upper bound is in very good agreement with the observational lower bound (from the RHIC facility). Furthermore, in a holographic approach using only Einstein gravity, η/s does not respond to variations of other physical parameters, while ν and Re can do so. In particular, it is known that the magnetic fields arising in peripheral heavy-ion collisions vary strongly with the impact parameter b, and we find that the holographic model predicts that ν and Re can also be expected to vary substantially with the magnetic field and therefore with b.

How does the Quark-Gluon Plasma know the collision energy?

“…This does not matter here since the temperature is fixed throughout our discussions.) From this we see that the kinematic viscosity is, according to holography, affected by the magnetic field in just the same way as s/ǫ: that is, these extremely intense fields reduce the momentum diffusion to a significant extent [39]. This can be expected to influence the jet quenching parameter, due to the effects mentioned earlier.…”

“…Gauge-gravity models of jet quenching are of course well known [35,36], and such models also exist which take into account the effect of the magnetic field [37]. Here we will use a much more basic "minimal" model [38,39] in which the bulk geometry is that of a magnetically charged dilatonic asymptotically AdS thermal black hole 3 . We stress again that there is no reason to expect, over the relatively narrow range of magnetic field values being considered here, any close agreement with the findings of [6][7][8]; indeed it is not clear that the holographically computed values of K will even change in the correct direction (smaller for the LHC plasmas, by a factor of around 2 or 3, or perhaps a little more for the 5.02 TeV collisions).…”

“…For the purposes of this work, the reader may regard the presence of the dilaton as essentially a technical matter, required (when magnetic fields are extremely strong relative to the squared temperature) in order to ensure that the bulk system is mathematically consistent within string theory 5 . The form taken by this mathematical consistency condition in the present application is discussed in detail in [46,38,39]. The value of the coupling α is chosen to be the minimal value that results in this condition being satisfied.…”

“…As explained above, this means that we will consider three values for eB ∞ : eB ∞ ≈ 60 m 2 π , eB ∞ ≈ 6 m 2 π , and eB ∞ ≈ 0.6 m 2 π . These data allow us to compute α ≈ 0.34, α ≈ 0, α ≈ 0 for the three respective cases 7 , after the manner of [38,39]. We can now solve (9), (10), and (11) for P * , r h , and M * in each of the three cases, and again equation (12) yields three values for s P LHC /ǫ; note that, by construction of course, ǫ is the same throughout these computations.…”

Viscosity vs. vorticity in the quark-gluon plasma

“…One can argue[24] that the proper measure of "viscosity" is a dimensionless quantity, the Reynolds number Re (which is small for very viscous fluids): it takes into account the size, density, and characteristic velocity of the system, as well as η; Re gives, for example, a way of assessing whether turbulence is likely to develop. (See[25] for a study of the Reynolds number of the QGP.)…”