Relativistic Dissipative Fluid Dynamics Research Articles

Theories of Relativistic Dissipative Fluid Dynamics.

Relativistic dissipative fluid dynamics finds widespread applications in high-energy nuclear physics and astrophysics. However, formulating a causal and stable theory of relativistic dissipative fluid dynamics is far from trivial; efforts to accomplish this reach back more than 50 years. In this review, we give an overview of the field and attempt a comparative assessment of (at least most of) the theories for relativistic dissipative fluid dynamics proposed until today and used in applications.

Third-order relativistic dissipative fluid dynamics from the method of moments

Third-order relativistic dissipative fluid dynamics from the method of moments

A causal formulation of dissipative relativistic fluid dynamics with or without diffusion

The article proposes a causal five-field formulation of dissipative relativistic fluid dynamics as a quasilinear symmetric hyperbolic system of second order. The system is determined by four dissipation coefficients η , ζ , κ , μ \eta ,\zeta ,\kappa ,\mu , free functions of the fields, which quantify shear viscosity, bulk viscosity, heat conductivity, and diffusion.

Multicomponent relativistic dissipative fluid dynamics from the Boltzmann equation

We derive multicomponent relativistic second-order dissipative fluid dynamics from the Boltzmann equations for a reactive mixture of ${N}_{\mathrm{spec}}$ particle species with ${N}_{q}$ intrinsic quantum numbers (e.g., electric charge, baryon number, and strangeness) using the method of moments. We obtain the continuity equations for multiple conserved charges as well as the conservation equations for the total energy and momentum in the single-fluid approximation. These $4+{N}_{q}$ conservation laws are closed by deriving the second-order equations of motion for the dissipative quantities in the $(10+4{N}_{q})$-moment approximation. The resulting fluid-dynamical equations are formally similar to those of a single-component system, but feature different thermodynamic relations and transport coefficients. We derive general relations for all transport coefficients and compute them explicitly in the ultrarelativistic limit.

Linear causality and stability of third-order relativistic dissipative fluid dynamics

We analyze the linear causality and stability of third-order fluid dynamics considering perturbations around a global equilibrium state. We investigate the formulation derived from kinetic theory, using the Chapman-Enskog expansion, in [Phys. Rev. C 88, 021903 (2013)] which was shown to be in excellent agreement with solutions of the microscopic theory. From this analysis, we demonstrate that this theory is linearly acausal and unstable and that such instabilities cannot be corrected by tuning the transport coefficients. We then propose a modification of this theory, valid only in the linear regime, that can be constructed to be linearly causal and stable and obtain the conditions the transport coefficients must satisfy in order for this to be the case.

Decay and subluminality of modes of all wave numbers in the relativistic dynamics of viscous and heat conductive fluids

To further confirm the causality and stability of a second-order hyperbolic system of partial differential equations that models the relativistic dynamics of barotropic fluids with viscosity and heat conduction [H. Freistühler and B. Temple, J. Math. Phys. 59, 063101 (2018)], this paper studies the Fourier–Laplace modes of this system and shows that all such modes, relative to arbitrary Lorentz frames, (a) decay with increasing time and (b) travel at subluminal speeds. Stability is also shown for the related model of non-barotropic fluids [H. Freistühler and B. Temple, Proc. R. Soc. A 470, 20140055 (2014) and H. Freistühler and B. Temple, Proc. R. Soc. A 473, 20160729 (2017)]. Even though these properties had been known for a while in the sense of numerical evidence, the fully analytical proofs for the subluminality of modes of arbitrary wave numbers in arbitrary frames given here appear to be the first regarding any four-field formulation of dissipative relativistic fluid dynamics.

From chiral kinetic theory to relativistic viscous spin hydrodynamics

In this work, we start with chiral kinetic theory and construct the spin hydrodynamic framework for a chiral spinor system. Using the 14-moment expansion formalism, we obtain the equations of motion of second-order dissipative relativistic fluid dynamics with non-trivial spin polarization density. In a chiral spinor system, the spin alignment effect could be treated in the same framework as the Chiral Vortical Effect (CVE). However, the quantum corrections due to fluid vorticity induce not only CVE terms in the vector/axial charge currents but also corrections to the stress tensor. In this framework, viscous corrections to the hadron spin polarization are self-consistently obtained, which will be important for the precise prediction of the polarization rate for the observed hadrons, e.g. $\Lambda$-hyperon.

From Chiral Kinetic Theory To Spin Hydrodynamics

The total angular momentum is conserved in the evolution of the Quark-Gluon Plasma (QGP) created in heavy-ion collision, and consists of two sectors: the orbital angular momentum (OAM) caused by kinetic motion, and the spin, an intrinsic degree of freedom of quarks and gluons. Microscopic scattering processes allow the conversion between these two components, hence the spin density could eventually have non-trivial influence on the QGP evolution. A hydrodynamic theory, with the spin polarization effect properly taken into account, is required to quantitatively study the polarization rate for observed hadrons, e.g. the Λ-hyperon. In this work, we start with chiral kinetic theory and construct the spin hydrodynamic framework for a chiral spinor system. We obtain the equations of motion of second-order dissipative relativistic fluid dynamics with non-trivial spin polarization density.

Dissipative hydrodynamics of relativistic shock waves in a quark gluon plasma: Comparing and benchmarking alternate numerical methods

This paper presents numerical cross-comparisons and benchmark results for two different kinetic numerical methods, capable of describing relativistic dissipative fluid dynamics in a wide range of kinematic regimes, typical of relevant physics applications, such as transport phenomena in quark-gluon plasmas. We refer to relativistic lattice Boltzmann versus Montecarlo Test-Particle methods. Lacking any realistic option for accurate validation vis-a-vis experimental data, we check the consistency of our results against established simulation packages available in the literature. We successfully cross-compare the results of the two aforementioned numerical approaches for momentum integrated quantities like the hydrostatic and dynamical pressure profiles, the collective flow and the heat flux. These results corroborate the confidence on the robustness and correctness of these computational methods and on the accurate calibration of their numerical parameters with respect to the physical transport coefficients. Our numerical results are made available as supplemental material, with the aim of establishing a reference benchmark for other numerical approaches.

A class of Hadamard well-posed five-field theories of dissipative relativistic fluid dynamics

This paper proposes a class of causal formulations for dissipative relativistic fluid dynamics as a hyperbolic five-field system of second-order partial differential equations. The argumentation continues lines of thinking developed in the author’s earlier papers with Temple and generalizes a four-field description that was recently given by Bemfica, Disconzi, and Noronha [Phys. Rev. D 98, 104064 (2018)] for the case of the pure radiation fluid. The Navier–Stokes–Fourier-like models are constructed as perturbations of auxiliary “involutory augmentations” of the Euler equations and shown to be Hadamard well-posed at least at the level of their linearization.

(3+1)-dimensional dissipative relativistic fluid dynamics at non-zero net baryon density

Heavy-ion collisions at center-of-mass energies between 1 and 100 GeV/nucleon are essential to understand the phase diagram of QCD and search for its critical point. At these energies the net baryon density of the system can be high, and simulating its evolution becomes an indispensable part of theoretical modeling. We here present the (3+1)-dimensional diffusive relativistic hydrodynamic code BEShydro which solves the equations of motion of second-order Denicol–Niemi–Molnar–Rischke (DNMR) theory, including bulk and shear viscous currents and baryon diffusion currents. BEShydro features a modular structure that allows to easily turn on and off baryon evolution and different dissipative effects and thus to study their physical effects on the dynamical evolution individually. An extensive set of test protocols for the code, including several novel tests of the precision of baryon transport that can also be used to test other such codes, is documented here and supplied as a permanent part of the code package. Program summaryProgram Title:BEShydroProgram Files doi:http://dx.doi.org/10.17632/xwywvb2psm.1Licensing provisions: GPLv3Programming language: C++External routines/libraries: GNU Scientific Library (GSL)Nature of problem: (3+1)-dimensional dynamical evolution of hot and dense matter created in relativistic heavy-ion collisions using second-order dissipative relativistic fluid dynamics, including evolution of net baryon number and its dissipative diffusion current.Solution method: Runge–Kutta Kurganov–Tadmor algorithm.

Asymptotic stability in a second-order symmetric hyperbolic system modeling the relativistic dynamics of viscous heat-conductive fluids with diffusion

This paper establishes nonlinear asymptotic stability of homogeneous reference states in dissipative relativistic fluid dynamics. The result is a counterpart for general non-barotropic fluids of one obtained by the author in a previous paper on barotropic fluids. Differently from that of this earlier finding, the proof here crucially relies on analyzing the corresponding linearized problem in Fourier space, with different scalings for small and large wave numbers.

Causality of fluid dynamics for high-energy nuclear collisions

Dissipative relativistic fluid dynamics is not always causal and can favor super-luminal signal propagation under certain circumstances. On the other hand, high-energy nuclear collisions have a microscopic description in terms of QCD and are expected to follow the causality principle of special relativity. We discuss under which conditions the fluid evolution equations for a radial expansion are hyperbolic and that terms of second order in the Knudsen number are problematic for causality. We also outline briefly how this can be remedied with terms of higher order in a formal derivative expansion. The expansion dynamics are causal in the relativistic sense if the characteristic velocities are smaller than the speed of light. We obtain a concrete inequality from this constraint and discuss how it can be violated for certain initial conditions. Finally we argue that causality poses a bound on the applicability of relativistic fluid dynamics.

Relativistic Dissipative Fluid Dynamics from the Non-Equilibrium Statistical Operator

We present a new derivation of second-order relativistic dissipative fluid dynamics for quantum systems using Zubarev’s formalism for the non-equilibrium statistical operator. In particular, we discuss the shear-stress tensor to second order in gradients and argue that the relaxation terms for the dissipative quantities arise from memory effects contained in the statistical operator. We also identify new transport coefficients which describe the relaxation of dissipative processes to second order and express them in terms of equilibrium correlation functions, thus establishing Kubo-type formulae for the second-order transport coefficients.

Causal dissipation for the relativistic dynamics of ideal gases.

We derive a general class of relativistic dissipation tensors by requiring that, combined with the relativistic Euler equations, they form a second-order system of partial differential equations which is symmetric hyperbolic in a second-order sense when written in the natural Godunov variables that make the Euler equations symmetric hyperbolic in the first-order sense. We show that this class contains a unique element representing a causal formulation of relativistic dissipative fluid dynamics which (i) is equivalent to the classical descriptions by Eckart and Landau to first order in the coefficients of viscosity and heat conduction and (ii) has its signal speeds bounded sharply by the speed of light. Based on these properties, we propose this system as a natural candidate for the relativistic counterpart of the classical Navier-Stokes equations.

Relativistic second-order dissipative fluid dynamics at finite chemical potential

We employ a Chapman-Enskog like expansion for the distribution function close to equilibrium to solve the Boltzmann equation in the relaxation time approximation and subsequently derive second-order evolution equations for dissipative charge currentand shear stress tensor for a system of massless quarks and gluons. We use quantum statistics for the phase space distribution functions to calculate the transport coefficients. We show that, the second-order evolution equations for the dissipative charge current and the shear stress tensor can be decoupled. We find that, for large chemical potential, the charge conductivity is small compared to the shear viscosity. Moreover, we demonstrate that the limiting behaviour of the ratio of heat conductivity to shear viscosity is identicalto that obtained for a strongly coupled conformal plasma.

Kinetic foundations of relativistic dissipative fluid dynamics

In this contribution we discuss in detail the most widespread formalisms employed to derive relativistic dissipative fluid dynamics from the Boltzmann equation: Chapman–Enskog expansion and Israel–Stewart theory. We further point out the drawbacks of each theory and explain possible ways to circumvent them. Recent developments in the derivation of fluid dynamics from the Boltzmann equation are also discussed.

Relative importance of second-order terms in relativistic dissipative fluid dynamics

In Denicol et al. [Phys. Rev. D 85, 114047 (2012)], the equations of motion of relativistic dissipative fluid dynamics were derived from the relativistic Boltzmann equation. These equations contain a multitude of terms of second order in the Knudsen number, in the inverse Reynolds number, or their product. Terms of second order in the Knudsen number give rise to nonhyperbolic (and thus acausal) behavior and must be neglected in (numerical) solutions of relativistic dissipative fluid dynamics. The coefficients of the terms which are of the order of the product of Knudsen and inverse Reynolds numbers have been explicitly computed in the above reference, in the limit of a massless Boltzmann gas. Terms of second order in the inverse Reynolds number arise from the collision term in the Boltzmann equation, upon expansion to second order in deviations from the single-particle distribution function in local thermodynamical equilibrium. In this work, we compute these second-order terms for a massless Boltzmann gas with constant scattering cross section. Consequently, we assess their relative importance in comparison to the terms which are of the order of the product of the Knudsen and inverse Reynolds numbers.

Solving the heat-flow problem with transient relativistic fluid dynamics

Israel-Stewart theory is a causal, stable formulation of relativistic dissipative fluid dynamics. This theory has been shown to give a decent description of the dynamical behavior of a relativistic fluid in cases where shear stress becomes important. In principle, it should also be applicable to situations where heat flow becomes important. However, it has been shown that there are cases where Israel-Stewart theory cannot reproduce phenomena associated with heat flow. In this paper, we derive a relativistic dissipative fluid-dynamical theory from kinetic theory which provides a good description of all dissipative phenomena, including heat flow. We explicitly demonstrate this by comparing this theory with numerical solutions of the relativistic Boltzmann equation.

Event-by-event distributions of azimuthal asymmetries in ultrarelativistic heavy-ion collisions

Relativistic dissipative fluid dynamics is a common tool to describe the space-time evolution of the strongly interacting matter created in ultrarelativistic heavy-ion collisions. For a proper comparison to experimental data, fluid-dynamical calculations have to be performed on an event-by-event basis. Therefore, fluid dynamics should be able to reproduce, not only the event-averaged momentum anisotropies, $<v_{n}>$, but also their distributions. In this paper, we investigate the event-by-event distributions of the initial-state and momentum anisotropies $\epsilon_n$ and $v_n$, and their correlations. We demonstrate that the event-by-event distributions of relative $v_n$ fluctuations are almost equal to the event-by-event distributions of corresponding $\epsilon_n$ fluctuations, allowing experimental determination of the relative anisotropy fluctuations of the initial state. Furthermore, the correlation $c(v_2,v_4)$ turns out to be sensitive to the viscosity of the fluid providing an additional constraint to the properties of the strongly interacting matter.