We show that the centrality and system-size dependence of elliptic flow measured at RHIC are fully described by a simple model based on eccentricity scaling and incomplete thermalization. We argue that the elliptic flow is at least 25% below the (ideal) ``hydrodynamic limit'', even for the most central Au-Au collisions. This lack of perfect equilibration allows for estimates of the effective parton cross section in the Quark-Gluon Plasma and of its viscosity to entropy density ratio. We also show how the initial conditions affect the transport coefficients and thermodynamic quantities extracted from the data, in particular the viscosity and the speed of sound.
In the Israel-Stewart's theory of second order hydrodynamics, we have analysed the recent PHENIX data on charged particles elliptic flow in Au+Au collisions. PHENIX data demand more viscous fluid in peripheral collisions than in central collisions. Over a broad range of collision centrality (0-10%- 50-60%), viscosity to entropy ratio ($η/s$) varies between 0-0.17.
In this work, we study the centrality as well as transverse momentum dependence of the dihadron correlation for Au+Au collisions at 200A GeV. The numerical simulations are carried out by using a hydrodynamical code NeXSPheRIO, where the initial conditions are obtained from a Regge-Gribov based microscopic model, NeXuS. In our calculations, the centrality windows are evaluated regarding multiplicity. The final correlations are obtained by the background subtraction via ZYAM methods, where higher harmonics are also considered explicitly. The correlations are evaluated for the 0 - 20\%, 20\%-40\%…
Event by event EPOS calculations in which the expansion of the system is described by {\it ideal} hydrodynamics reproduce well the measured centrality dependence of $v_2/ε_{part}$, although it has been claimed that only viscous hydrodynamics can reproduce these data. This is due to the core - corona effect which manifests itself in the initial condition of the hydrodynamical expansion. The centrality dependence of $v_2/ε_{part}$ can be understood in the recently advanced core-corona model, a simple parameter free EPOS inspired model to describe the centrality dependence of different observable…
Measurements of di-hadron azimuthal correlations at different centralities for Au+Au collisions at 200 AGeV were reported by the PHENIX Collaboration. The data were presented for different ranges of transverse momentum. In particular, it was observed that the away-side correlation evolves from double- to a single-peak structure when the centrality decreases. In this work, we show that these features naturally appear as due to an interplay between the centrality-dependent smooth background elliptic flow and the one produced by event-by-event fluctuating peripheral tubes. To compare with the PHE…
In Israel-Stewart's theory of dissipative hydrodynamics, we have analyzed the recent ALICE data for the centrality dependence of charged particle multiplicity, centrality dependence of integrated elliptic flow and 0-5% charged particles $p_T$ spectra in $\sqrt{s}_{NN}$=2.76 TeV Pb+Pb collisions and determined the initial or the thermalisation time. Analysis indicate that the ALICE data disfavor very early thermalsiation $τ_i$=0.2 fm, or very late thermalsation $τ_i$= 4fm. Data are best explained for thermalisation time $τ_i=0.88^{-0.14}_{+0.68}$ fm.
Bulk viscosity suppresses elliptic flow v_2, as does shear viscosity. It can thus not be neglected when extracting the shear viscosity from elliptic flow data. We here explore uncertainties in the bulk viscous contribution to viscous v_2 suppression that arise from presently uncontrolled uncertainties in the initial value of the bulk viscous pressure and its microscopic relaxation time.
We investigate the interplay between hydrodynamic fluctuations and initial longitudinal fluctuations for their effects on the rapidity decorrelation of collective flow in high-energy nuclear collisions. We use a (3+1)-dimensional integrated dynamical model in which we combine initial conditions with longitudinal fluctuations, fluctuating hydrodynamics and hadronic cascades. We analyse the factorisation ratio in the longitudinal direction to study the effect of these fluctuations on the rapidity decorrelation. We find an essential difference between the effects of the hydrodynamic fluctuations …
In our analysis, we combine event-by-event hydrodynamics, within the EKRT formulation, with jet quenching -ASW Quenching Weights- to obtain high-$p_T$ $R_{\rm AA}$ for charged particles at RHIC and LHC energies for different centralities. By defining a $K$-factor that quantifies the departure of $\hat{q}$ from an ideal estimate, $K = \hat{q}/(2ε^{3/4})$, we fit the single-inclusive experimental data for charged particles. This $K$-factor is larger at RHIC than at the LHC but, surprisingly, it is almost independent of the centrality of the collision.
The particle spectra and Hanbury-Brown Twiss (HBT) radius of Au+Au collisions at RHIC energy are investigated by a hydrodynamical expanding source with both shear and bulk viscosities ($ζ$). With a large width of the ratio of $ζ$ to entropy density $s$, both the particle transverse momentum spectra and the ratio $R_{out}/R_{side}$ of HBT radii in the direction of the total transverse momentum of detected two particles ($R_{out}$) and perpendicular to both this direction and the beam direction ($R_{side}$) become a little steeper.
We study the influence of a temperature-dependent shear viscosity over entropy density ratio $η/s$, different shear relaxation times $τ_π$, as well as different initial conditions on the transverse momentum spectra of charged hadrons and identified particles. We investigate the azimuthal flow asymmetries as a function of both collision energy and centrality. The elliptic flow coefficient turns out to be dominated by the hadronic viscosity at RHIC energies. Only at higher collision energies the impact of the viscosity in the QGP phase is visible in the flow asymmetries. Nevertheless, the shear …
Effect of viscosity on one dimensional hydrodynamic flow and direct photons from 200 AGeV S+Au collisions at CERN SPS
我们在一维流体动力学模型中分析了CERN SPS上WA 80合作在200 A GeV S + Au碰撞中获得的直接光子数据。考虑了两种情况：（i）形成夸克-胶子等离子体和（ii）形成热强子气体。对于这两种情况，考虑了理想的以及极端粘性的流体。结果表明，无论流体是理想流体还是极端粘性流体，对QGP的直接光子产额影响不大。然而，如果产生强子气体，产率则有很大的不同。这两种情况都没有对数据作出令人满意的描述。
As one of the possible signals for the whereabouts of the critical point on the QCD phase diagram, recently, the multiplicity fluctuations in heavy-ion collisions have aroused much attention. It is a crucial observable of the Beam Energy Scan program of the Relativistic Heavy Ion Collider. In this work, we investigate the centrality dependence of the multiplicity fluctuations regarding the recent measurements from STAR Collaboration. By employing a hydrodynamical approach, the present study is dedicated to the noncritical aspects of the phenomenon. To be specific, in addition to the thermal fl…
The fourth harmonic of the azimuthal distribution of particles $v_4$ has been measured for Au-Au collisions at the Relativistic Heavy Ion Collider (RHIC). The centrality dependence of $v_4$ does not agree with the prediction from hydrodynamics. In particular, the ratio $v_4/(v_2)^2$, where $v_2$ denotes the second harmonic of the azimuthal distribution of particles, is significantly larger than predicted by hydrodynamics. We argue that this discrepancy is mostly due to elliptic flow ($v_2$) fluctuations. We evaluate these fluctuations on the basis of a Monte Carlo Glauber calculation. The effe…
We have studied the effect of nonzero bulk viscosity with peak near the lattice QCD predicted crossover temperature $T_{co}\sim 175 MeV$ on charged particle transverse momentum spectra and elliptic flow. The Israel-Stewart theory of 2nd order causal dissipative relativistic fluid dynamics is used to simulate the space time evolution of the matter formed in Au-Au collisions at $\sqrt{s_{NN}}$=200 GeV assuming longitudinal boost invariance. A systematic comparison of temperature, transverse velocity, spatial and momentum anisotropy evolution of the ideal, bulk and shear viscous fluid has been ca…
Relativistic heavy ion collisions generate nuclear-sized droplets of quark-gluon plasma (QGP) that exhibit nearly inviscid hydrodynamic expansion. Smaller collision systems such as p+Au, d+Au, and $^{3}$He+Au at the Relativistic Heavy Ion Collider, as well as p+Pb and high-multiplicity p+p at the Large Hadron Collider may create even smaller droplets of QGP. If so, the standard time evolution paradigm of heavy ion collisions may be extended to these smaller systems. These small systems present a unique opportunity to examine pre-hydrodynamic physics and extract properties of the QGP, such as t…
We study evolution of quark-gluon matter in the ultrarelativistic heavy-ion collisions within the frame work of relativistic second-order viscous hydrodynamics. In particular, by using the various prescriptions of a temperature-dependent shear viscosity to the entropy ratio, we show that the hydrodynamic description of the relativistic fluid become invalid due to the phenomenon of cavitation. For most of the initial conditions relevant for LHC, the cavitation sets in very early during the evolution of the hydrodynamics in time $\lesssim 2 $fm/c. The cavitation in this case is entirely driven b…
One of the most striking results from the Relativistic Heavy Ion Collider is the strong elliptic flow. This review summarizes what is observed and how these results are combined with reasonable theoretical assumptions to estimate the shear viscosity of QCD near the phase transition. A data comparison with viscous hydrodynamics and kinetic theory calculations indicates that the shear viscosity to entropy ratio is surprisingly small, $η/s < 0.4$. The preferred range is $η/s \simeq (1\leftrightarrow 3) \times 1/4π$.
We briefly discuss the recent developments in causal dissipative hydrodynamic for relativistic heavy ion collisions. Phenomenological estimate of QGP viscosity over entropy ratio from several experimental data, e.g. STAR's $ϕ$ meson data, centrality dependence of elliptic flow, universal scaling elliptic flow etc. are discussed. QGP viscosity, extracted from hydrodynamical model analysis can have very large systematic uncertainty due to uncertain initial conditions.
In this article, we will briefly review the recent progress on hydrodynamic modeling and the extraction of the quark-gluon plasma (QGP) specific shear viscosity with an emphasis on results obtained from the hybrid model VISHNU that couples viscous hydrodynamics for the macroscopic expansion of the QGP to the hadron cascade model for the microscopic evolution of the late hadronic stage.
Hydro-kinetic theory of thermal fluctuations is applied to a non-conformal relativistic fluid. Solving the hydro-kinetic equations for an isotropically expanding background we find that hydrodynamic fluctuations give ultraviolet divergent contributions to the energy-momentum tensor. After shifting the temperature to account for the energy of non-equilibrium modes, the remaining divergences are renormalized into local parameters, e.g. pressure and bulk viscosity. We also confirm that the renormalization of the pressure and bulk viscosity is universal by computing them for a Bjorken expansion. T…
We confirm and expand on work by Chafin and Schaefer on hydrodynamic fluctuations in the unitary Fermi gas. Using the result for the equation of state from a recent MIT experiment, we derive lower bounds for η/n and η/s as a function of temperature. Re-analyzing recent quantum Monte Carlo data for the shear-viscosity spectral function we point out a possible resolution for the tension between the viscosity bound η/n> 0.3 from Chafin and Schaefer and the quantum Monte Carlo results η/n<0.2$ from Wlazlowski et al. near the critical temperature
Hydrodynamics predicts long-lived sound and shear waves. Thermal fluctuations in these waves can lead to the diffusion of momentum density, contributing to the shear viscosity and other transport coefficients. Within viscous hydrodynamics in 3+1 dimensions, this leads to a positive contribution to the shear viscosity, which is finite but inversely proportional to the microscopic shear viscosity. Therefore the effective infrared viscosity is bounded from below. The contribution to the second-order transport coefficient $τ_π$ is divergent, which means that second-order relativistic viscous hydro…
We determine the shear viscosity of the ultracold Fermi gas at unitarity in the normal phase using hydrodynamic expansion data. The analysis is based on a generalized fluid dynamic framework which ensures a smooth transition between the fluid dynamic core of the cloud and the ballistic corona. We use expansion data taken by Joseph et al. and measurements of the equation of state by Ku et al. We find that the shear viscosity to particle density ratio just above the critical temperature is $η/n|_{T_c}=0.41\pm 0.11$. We also obtain evidence that the shear viscosity to entropy density ratio has a …
Taking into account of entropy generation during evolution of a viscous fluid, we have estimated inverse Knudsen number, ideal hydrodynamic limit for elliptic flow and QGP viscosity to entropy ratio in $\sqrt{s}$=62 and 200 GeV Cu+Cu/Au+Au collisions. Viscosity to entropy ratio is estimated as $η/s=0.17\pm 0.10\pm 0.20$, the first error is statistical, the second one is systematic. In a central Au+Au collision, inverse Knudsen number is $\approx 2.80\pm 1.63$, which presumably small for complete equilibration. In peripheral collisions it is even less. Ideal hydrodynamic limit for elliptic flow…
Bulk viscosity effects on the collective flow harmonics in heavy ion collisions are investigated, on an event by event basis, using a newly developed 2+1 Lagrangian hydrodynamic code named v-USPhydro which implements the Smoothed Particle Hydrodynamics (SPH) algorithm for viscous hydrodynamics. A new formula for the bulk viscous corrections present in the distribution function at freeze-out is derived starting from the Boltzmann equation for multi-hadron species. Bulk viscosity is shown to enhance the collective flow Fourier coefficients from $v_2(p_T)$ to $v_5(p_T)$ when $% p_{T}\sim 1-3$ GeV…
In the Israel-Stewart's theory of second order hydrodynamics, we have simulated $\sqrt{s}$=2.76 TeV Pb+Pb collisions. ALICE data for the centrality dependence of charged particles multiplicity, $p_T$ spectra in 0-5% collisions, centrality dependence of integrated and differential elliptic flow are analysed. Analysis indicate that while ALICE data on charged particles multiplicity or $p_T$ spectra do not demand any viscosity, viscosity is demanded by the elliptic flow data. From a simulataneous fit to all the data sets, viscosity to entropy ratio in $\sqrt{s}_{NN}$=2.76 TeV Pb+Pb collisions is …
With the PICR hydrodynamic model, we study the polarization splitting between $Λ$ and $\barΛ$ at RHIC BES energy range, based on the meson field mechanism. Our results fit to the experimental data fairly well. Besides, two unexpected effect emerges: (1) the baryon density gradient has non-trivial and negative contribution to the polarization splitting; (2) for 7.7 GeV Au+Au collisions within the centrality range of 20\%-50\%, the polarization splitting surprisingly increases with the centrality decreases. The second effect might help to explain the significant signal of polarization splitting …
Within Israel-Stewart's theory of dissipative hydrodynamics, we have analyzed ALICE data for the centrality dependence of charged particles multiplicity, elliptic flow and $p_T$ spectra in $\sqrt{s}_{NN}$=2.76 TeV Pb+Pb collisions and obtained the temperature dependence of the QGP viscosity over the entropy ratio ($\etas$). If temperature dependence of $\etas$ is parameterized as $\etas=α\frac{T-T_c}{T_c}+\frac{1}{4π}$, experimental data favor $α$ in the range 0-0.2. $α\geq 0.4$ is not favored by the data. Experimental data in $\sqrt{s}_{NN}$=200 GeV Au+Au collisions however, prefer $α$=0.4.
The interplay between shear and bulk viscosities on the flow harmonics, $v_n$'s, at RHIC is investigated using the newly developed relativistic 2+1 hydrodynamical code v-USPhydro that includes bulk and shear viscosity effects both in the hydrodynamic evolution and also at freeze-out. While shear viscosity is known to attenuate the flow harmonics, we find that the inclusion of bulk viscosity decreases the shear viscosity-induced suppression of the flow harmonics bringing them closer to their values in ideal hydrodynamical calculations. Depending on the value of the bulk viscosity to entropy den…
We investigate the implications of a nonzero bulk viscosity coefficient on the azimuthal momentum anisotropy of ultracentral relativistic heavy ion collisions at the Large Hadron Collider. We find that, with IP-Glasma initial conditions, a finite bulk viscosity coefficient leads to a better description of the flow harmonics in ultracentral collisions. We then extract optimal values of bulk and shear viscosity coefficients that provide the best agreement with flow harmonic coefficients data in this centrality class.
Fluctuations in the initial QCD matter density distribution are found to enhance the production of thermal photons significantly in the range 2 \leq pT \leq 4 GeV/c compared to a smooth initial state averaged profile in ideal hydrodynamic calculation for 200 AGeV Au+Au collisions at the Relativistic Heavy Ion Collider (RHIC) and 2.76 ATeV Pb+Pb collisions at the Large Hadron Collider (LHC). The thermal emission of photons is strongly dependent on the initial temperature of the system where the presence of 'hotspots' in the initial state translates into enhanced production of photons compared t…
We compute the initial energy densities produced in ultrarelativistic heavy-ion collisions from NLO perturbative QCD using a saturation conjecture to control soft particle production, and describe the subsequent space-time evolution of the system with hydrodynamics, event by event. The resulting centrality dependence of the low-$p_T$ observables from this pQCD + saturation + hydro ("EKRT") framework are then compared simultaneously to the LHC and RHIC measurements. With such an analysis we can test the initial state calculation, and constrain the temperature dependence of the shear viscosity-t…
We study hydrodynamic fluctuations in a non-relativistic fluid. We show that in three dimensions fluctuations lead to a minimum in the shear viscosity to entropy density ratio $η/s$ as a function of the temperature. The minimum provides a bound on $η/s$ which is independent of the conjectured bound in string theory, $η/s \geq \hbar/(4πk_B)$, where $s$ is the entropy density. For the dilute Fermi gas at unitarity we find $η/s\gsim 0.2\hbar$. This bound is not universal -- it depends on thermodynamic properties of the unitary Fermi gas, and on empirical information about the range of validity of…
We solve second order relativistic hydrodynamics equations for a boost-invariant 1+1-dimensional expanding fluid with an equation of state taken from lattice calculations of the thermodynamics of strongly coupled quark-gluon plasma. We investigate the dependence of the energy density as a function of proper time on the values of the shear viscosity, the bulk viscosity, and second order coefficients, confirming that large changes in the values of the latter have negligible effects. Varying the shear viscosity between zero and a few times s/(4 pi), with s the entropy density, has significant eff…
The deuteron binding energy is only 2.2 MeV. At the same time, its yield in Pb+Pb collisions at $\sqrt{s_{NN}} = $2.76 TeV corresponds to a thermal yield at the temperature around 155 MeV, which is too hot to keep deuterons bound. This puzzle is not completely resolved yet. In general, the mechanism of light nuclei production in ultra-high energy heavy ion collisions remains under debate. In a previous work [1] we suggest a microscopic explanation of the deuteron production in central ultra-relativistic Pb+Pb collisions, the main mechanism being $πpn \leftrightarrow πd$ reactions in the hadron…
Centrality dependence of charged particles multiplicity, transverse momentum spectra, integrated and differential elliptic flow, in $\sqrt{s}_{NN}$=200 GeV Au+Au collisions are analyzed using event by event ideal hydrodynamics. Monte-Carlo Glauber model of initial condition, constrained to reproduce experimental charged particle's multiplicity in 0-10% Au+Au collisions, reasonably well reproduces all the experimental observables, e.g. centrality dependence of charged particles multiplicity, integrated and differential elliptic flow. Model predictions for higher flow harmonics, $v_3$, $v_4$ how…
One of the most remarkable features of the Quark Gluon Plasma is its nearly perfect fluidity behavior indicated by the small shear viscosity to entropy density ratio obtained from fitting relativistic viscous hydrodynamics flow harmonics to experimental data. In recent years, bulk viscosity has also been considered in the context of event-by-event relativistic hydrodynamics and it has been found to have a non-trivial interplay with shear viscosity. In this paper some of the issues are discussed that require further work when extracting the shear viscosity to entropy density ratio in the presen…
The space-time evolution of the (1+1)-dimensional viscous hydrodynamics with an initial quark-gluon plasma (QGP) produced in ultrarelativistic heavy ion collisions is studied numerically. The particle-emitting sources undergo a crossover transition from the QGP to hadronic gas. We take into account a usual shear viscosity for the strongly coupled QGP as well as the bulk viscosity which increases significantly in the crossover region. The two-pion Hanbury-Brown-Twiss (HBT) interferometry for the viscous hydrodynamic sources is performed. The HBT analyses indicate that the viscosity effect on th…
Charged particles $p_T$ spectra and elliptic flow in 0-60% Au+Au collisions at RHIC are analyzed in a hydrodynamic model with hot, hadronic resonance gas in the initial state. Physically conceivable hadronic resonance gas, thermalized in the time scale $τ_i$=1 fm, at a (central) temperature $T_i$=220 MeV, with viscosity to entropy ratio $η/s$=0.24, reasonably well explains $p_T$ spectra in all the collision centralities. Centrality dependence of elliptic flow however demands continual increase of viscosity to entropy ratio with centrality.
Shear viscosity from fluctuations in linearized relativistic hydrodynamics
České vysoké učení technické v Praze（布拉格捷克理工大学）；SUBATECH UMR 6457 (IMT Atlantique, Université de Nantes, IN2P3/CNRS)（南特大学/大西洋高等理工学院/法国国家科学研究中心）；Univerzita Mateja Bela（马特伊·贝尔大学）
A new framework for evaluating hydrodynamic models of relativistic heavy ion collisions has been developed. This framework, a Comprehesive Heavy Ion Model Evaluation and Reporting Algorithm (CHIMERA) has been implemented by augmenting UVH 2+1D viscous hydrodynamic model with eccentricity fluctuations, pre-equilibrium flow, and the Ultra-relativistic Quantum Molecular Dynamic (UrQMD) hadronic cascade. A range of initial temperatures and shear viscosity to entropy ratios were evaluated for four initial profiles, $N_{part}$ and $N_{coll}$ scaling with and without pre-equilibrium flow. The model r…
We study the small frequency behavior of the bulk viscosity spectral function using stochastic fluid dynamics. We obtain a number of model independent results, including the long-time tail of the bulk stress correlation function, and the leading non-analyticity of the spectral function at small frequency. We also establish a lower bound on the bulk viscosity which is weakly dependent on assumptions regarding the range of applicability of fluid dynamics. The bound on the bulk viscosity $ζ$ scales as $ζ_{\it min} \sim (P-\frac{2}{3}{\cal E})^2 \sum_i D_i^{-2}$, where $D_i$ are the diffusion cons…
We study chiral magnetic effect in collisions of AuAu, RuRu and ZrZr at s = 200GeV. The axial charge evolution is modeled with stochastic hydrodynamics and geometrical quantities are calculated with Monte Carlo Glauber model. By adjusting the relaxation time of magnetic field, we find our results in good agreement with background subtracted data for AuAu collisions at the same energy. We also make prediction for RuRu and ZrZr collisions. We find a weak centrality dependence of initial chiral imbalance, which implies the centrality dependence of chiral magnetic effect signal comes mainly from t…
New, analytic solutions of relativistic viscous hydrodynamics are presented, describing expanding fireballs with Hubble-like velocity profile and ellipsoidal symmetry, similar to fireballs created in heavy ion collisions. We find that with these specifications, one obtains solutions where the shear viscosity essentially does not influence the time evolution of the system, thus these solutions are particularly adept tools to study the effect of bulk viscosity alone, which always results in a slower decrease of energy density as well as temperature compared to the case of perfect fluid. We inves…
Using the (2+1)-dimensional viscous hydrodynamic code VISH2+1 with a temperature dependent specific shear viscosity (eta/s)(T), we present a detailed study of the influence of a large hadronic shear viscosity and its corresponding relaxation time tau_pi on the transverse momentum spectra and elliptic flow of hadrons produced in 200 A GeV Au+Au collisions. Although theory, in principle, predicts a well-defined relation T*tau_pi = kappa(T)*(eta/s)(T), the precise form of kappa(T) for the matter created in relativistic heavy-ion collisions is not known. For the popular choice kappa=3 the hadron s…
We introduce an event-by-event pQCD + saturation + hydro ("EKRT") framework for high-energy heavy-ion collisions, where we compute the produced fluctuating QCD-matter energy densities from next-to-leading order (NLO) perturbative QCD (pQCD) using saturation to control soft particle production, and describe the space-time evolution of the QCD matter with viscous hydrodynamics, event by event (EbyE). We compare the computed centrality dependence of hadronic multiplicities, p_T spectra and flow coefficients v_n against LHC and RHIC data. We compare also the computed EbyE probability distributions…
We present event-by-event viscous hydrodynamic calculations of the anisotropic flow coefficients v_2 to v_5 for heavy-ion collisions at the Relativistic Heavy-Ion Collider (RHIC). We study the dependence of different flow harmonics on shear viscosity and the morphology of the initial state. v_3 and higher flow harmonics exhibit a particularly strong dependence on both the initial granularity and shear viscosity. We argue that a combined analysis of all available flow harmonics will allow to determine eta/s of the quark gluon plasma precisely. Presented results strongly hint at a value (eta/s)_…
A Fermi gas of atoms with resonant interactions is predicted to obey universal hydrodynamics, where the shear viscosity and other transport coefficients are universal functions of the density and temperature. At low temperatures, the viscosity has a universal quantum scale $\hbar n$ where $n$ is the density, while at high temperatures the natural scale is $p_T^3/\hbar^2$ where $p_T$ is the thermal momentum. We employ breathing mode damping to measure the shear viscosity at low temperature. At high temperature $T$, we employ anisotropic expansion of the cloud to find the viscosity, which exhibi…
