Israel-Stewart Hydrodynamics Research Articles

Regime of applicability of Israel-Stewart hydrodynamics

Regime of applicability of Israel-Stewart hydrodynamics

From moments of the distribution function to hydrodynamics: The nonconformal case

We study the one-dimensional boost-invariant Boltzmann equation in the relaxation-time approximation using special moments of the distribution function for a system with a finite particle mass. The infinite hierarchy of moments can be truncated by keeping only the three lowest moments that correspond to the three independent components of the energy-momentum tensor. We show that such a three-moment truncation reproduces accurately the exact solution of the kinetic equation after a simple renormalization that takes into account the effects of the neglected higher moments. We derive second-order Israel-Stewart hydrodynamic equations from the three-moment equations, and show that, for most physically relevant initial conditions, these equations yield results comparable to those of the three-moment truncation, albeit less accurate. We attribute this feature to the fact that the structure of Israel-Stewart equations is similar to that of the three-moment truncation. In particular, the presence of the relaxation term in the Israel-Stewart equations, yields an early-time regime that mimics approximately the collisionless regime. A detailed comparison of the three-moment truncation with second-order non-conformal hydrodynamics reveals ambiguities in the definition of second-order transport coefficients. These ambiguities affect the ability of Israel-Stewart hydrodynamics to reproduce results of kinetic theory.

Fluctuating relativistic dissipative hydrodynamics as a gauge theory

We argue that different formulations of hydrodynamics are related to uncertainties in the definitions of local thermodynamic and hydrodynamic variables. We show that this ambiguity can be resolved by viewing different formulations of hydrodynamics as particular “gauge choices” which lead to the same physical behavior of the system. Using the example of bulk viscosity, we show that Bemfica–Disconzi–Noronha–Kovtun (BDNK) and Israel–Stewart hydrodynamics are particular “gauge choices” of this type, related by a well-defined transformation of thermodynamic and hydrodynamic variables. We argue that this gauge ambiguity is necessary to ascertain the causality of stochastic hydrodynamic evolution and conjecture that it could explain the applicability of hydrodynamics outside its expected regime of validity since far from equilibrium and close to equilibrium may be related through transformations of this type.

Hybrid model with viscous relativistic hydrodynamics: a role of constraints on the shear-stress tensor

We present the hybrid hadron string dynamic (HydHSD) model connecting the parton-hadron-string dynamic model (PHSD) and a hydrodynamic model taking into account shear viscosity within the Israel-Stewart approach. The performance of the code is tested on the pion and proton rapidity and transverse mass distributions calculated for Au+Au and Pb+Pb collision at AGS--SPS energies. The influence of the switch time from transport to hydro models, the viscous parameter, and freeze-out time are discussed. Since the applicability of the Israel-Stewart hydrodynamics assumes the perturbative character of the viscous stress tensor, $\pi^{\mu\nu}$, which should not exceed the ideal energy-momentum tensor, $T_{\rm id}^{\mu\nu}$, hydrodynamical codes usually rescale the shear stress tensor if the inequality $\|\pi^{\mu\nu}\|\ll \|T_{\rm id}^{\mu\nu}\|$ is not fulfilled in some sense. We show that the form of the corresponding condition plays an important role in the sensitivity of hydrodynamic calculations to the viscous parameter -- a ratio of the shear viscosity to the entropy density, $\eta/s$. It is shown that the constraints used in the vHLLE and MUSIC models give the same results for the observables. With these constraints, the rapidity distributions and transverse momentum spectra are most sensitive to a change of the $\eta/s$ ratio. As an alternative, a strict condition is used. We performed global fits the rapidity and transverse mass distribution of pion and protons. It was also found that $\eta/s$ as a function of the collision energy monotonically increases from $E_{\rm lab}=6A$GeV up to $E_{\rm lab}=40A$GeV and saturates for higher SPS energies. We observe that it is difficult to reproduce simultaneously pion and proton rapidity distribution within our model with the present choice of the equation of state without a phase transition.

Exploring theoretical uncertainties in the hydrodynamic description of relativistic heavy-ion collisions

We explore theoretical uncertainties in the hydrodynamic description of relativistic heavy-ion collisions by examining the full non-linear causality conditions and quantifying the second-order transport coefficients' role on flow observables. The causality conditions impose physical constraints on the maximum allowed values of inverse Reynolds numbers during the hydrodynamic evolution. Including additional second-order gradient terms in the Denicol-Niemi-Moln\'{a}r-Rischke (DNMR) theory significantly shrinks the casual regions compared to those in the Israel-Stewart hydrodynamics. For Au+Au collisions, we find the variations of flow observables are small with and without imposing the necessary causality conditions, suggesting a robust extraction of the Quark-Gluon Plasma's transport coefficients in previous model-to-data comparisons. However, sizable sensitivity is present in small p+Au collisions, which poses challenges to study the small systems' collectivity.

Determining transport coefficients for a microscopic simulation of a hadron gas

Quark-Gluon plasmas produced in relativistic heavy-ion collisions quickly expand and cool, entering a phase consisting of multiple interacting hadronic resonances just below the QCD deconfinement temperature, $T\sim 155$ MeV. Numerical microscopic simulations have emerged as the principal method for modeling the behavior of the hadronic stage of heavy-ion collisions, but the transport properties that characterize these simulations are not well understood. Methods are presented here for extracting the shear viscosity, and two transport parameters that emerge in Israel-Stewart hydrodynamics. The analysis is based on studying how the stress-energy tensor responds to velocity gradients. Results agree with expectations based on Kubo relations.

Rapidity Correlation Structures from Causal Hydrodynamics

Viscous diffusion can broaden the rapidity dependence of two-particle transverse momentum fluctuations. Surprisingly, measurements at RHIC by the STAR collaboration demonstrate that this broadening is accompanied by the appearance of unanticipated structure in the rapidity distribution of these fluctuations in the most central collisions. Although a first order classical Navier-Stokes theory can roughly explain the rapidity broadening, it cannot explain the additional structure. We propose that the rapidity structure can be explained using the second order causal Israel-Stewart hydrodynamics with stochastic noise.

Nonlinear waves in second order conformal hydrodynamics

In this work we study wave propagation in dissipative relativistic fluids described by a simplified set of the 2nd order viscous conformal hydrodynamic equations corresponding to Israel–Stewart theory. Small amplitude waves are studied within the linearization approximation while waves with large amplitude are investigated using the reductive perturbation method, which is generalized to the case of 2nd order relativistic hydrodynamics. Our results indicate the presence of a “soliton-like” wave solution in Israel–Stewart hydrodynamics despite the presence of dissipation and relaxation effects.

Rapidity Dependence of Transverse Momentum Correlations from Fluctuating Hydrodynamics

Interest in the development of the theory of fluctuating hydrodynamics is growing [1]. Early efforts suggested that viscous diffusion broadens the rapidity dependence of two-particle transverse momentum correlations [2]. That work stimulated an experimental analysis by STAR [3]. In this work we present results of correlation observables computed using the second order Israel-Stewart hydrodynamics with stochastic noise and latest theoretical equations of state and transport coefficients. We also compute these observables using the first Navier-Stokes theory. We find that the second order theory better describes the data and also provides some novel features forced by causal constraints of the second order theory.

PHOTON EMISSION FROM OUT OF EQUILIBRIUM DISSIPATIVE PARTON PLASMA

Using the second order Israel–Stewart hydrodynamics we discuss the effect of viscosity on photon production in a parton plasma created in relativistic heavy ion collisions. We find that photon production rates can be enhanced by several factors due to the viscous effect in a chemically nonequilibrated plasma.

Shear viscosity and out of equilibrium dynamics

Using Grad's method, we calculate the entropy production and derive a formula for the second-order shear viscosity coefficient in a one-dimensionally expanding particle system, which can also be considered out of chemical equilibrium. For a one-dimensional expansion of gluon matter with Bjorken boost invariance, the shear tensor and the shear viscosity to entropy density ratio $\eta/s$ are numerically calculated by an iterative and self-consistent prescription within the second-order Israel-Stewart hydrodynamics and by a microscopic parton cascade transport theory. Compared with $\eta/s$ obtained using the Navier-Stokes approximation, the present result is about 20% larger at a QCD coupling $\alpha_s \sim 0.3$(with $\eta/s\approx 0.18$) and is a factor of 2-3 larger at a small coupling $\alpha_s \sim 0.01$. We demonstrate an agreement between the viscous hydrodynamic calculations and the microscopic transport results on $\eta/s$, except when employing a small $\alpha_s$. On the other hand, we demonstrate that for such small $\alpha_s$, the gluon system is far from kinetic and chemical equilibrium, which indicates the break down of second-order hydrodynamics because of the strong noneqilibrium evolution. In addition, for large $\alpha_s$ ($0.3-0.6$), the Israel-Stewart hydrodynamics formally breaks down at large momentum $p_T\gtrsim 3$ GeV but is still a reasonably good approximation.

Dissipative effects from transport and viscous hydrodynamics

We compare 2 → 2 covariant transport theory and causal Israel–Stewart hydrodynamics in a (2 + 1)-dimensional longitudinally boost-invariant geometry with RHIC-like initial conditions and a conformal ε = 3p equation of state. The pressure evolution in the center of the collision zone and the final differential elliptic flow v2(pT) from the two theories agree remarkably well for a small shear viscosity to entropy density ratio η/s ≈ 1/(4π), and also for a large cross section σ ≈ 50 mb. A key to this agreement is keeping all terms in the Israel–Stewart equations of motion. Our results indicate promising prospects for the applicability of Israel–Stewart dissipative hydrodynamics at RHIC, provided the shear viscosity of hot and dense quark–gluon matter is indeed very small for the relevant temperatures T ∼ 200–500 MeV.

Aspects of causal viscous hydrodynamics

We investigate the phenomenology of freely expanding fluids, with different material properties, evolving through the Israel-Stewart (IS) causal viscous hydrodynamics, and compare our results with those obtained in the relativistic Eckart-Landau-Navier-Stokes (ELNS) acausal viscous hydrodynamics. Through the analysis of scaling invariants we give a definition of thermalization time that can be self-consistently determined in viscous hydrodynamics. Next we construct the solutions for one-dimensional boost-invariant flows. Expansion of viscous fluids is slower than that of one-dimensional ideal fluids, resulting in entropy production. At late times, these flows are reasonably well approximated by solutions obtained in ELNS hydrodynamics. Estimates of initial energy densities from observed final values are strongly dependent on the dynamics one chooses. For the same material, and the same final state, IS hydrodynamics gives the smallest initial energy density. We also study fluctuations about these one-dimensional boost-invariant backgrounds; they are damped in ELNS hydrodynamics but can become sound waves in IS hydrodynamics. The difference is obvious in power spectra due to clear signals of wave-interference in IS hydrodynamics, which is completely absent in ELNS dynamics.