Relativistic Dissipative Fluid 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

Numerical simulations of divergence-type theories for conformal dissipative fluids

We present the first numerical simulations of the symmetric-hyperbolic theory for conformal dissipative relativistic fluids developed in Lehner et al. [Hyperbolic theory of relativistic conformal dissipative fluids, Phys. Rev. D 97, 024013 (2018)]. In this theory, the information of the fluid dynamics is encoded in a scalar generating function which depends on three free parameters. By adapting the WENO-Z high-resolution shock-capturing central scheme, we present numerical solutions restricted to planar symmetry in Minkowski spacetime, from two qualitatively different initial data: a smooth bump and a discontinuous step. We perform a detailed exploration of the effect of the different parameters of the theory, and numerically assess the constitutive relations associated with the shear viscosity by analyzing the entropy production rate when shocks are produced.

Linear response hydrodynamics of a relativistic dissipative fluid with spin

We formulate a Lagrangian hydrodynamics including shear and bulk viscosity in the presence of spin density, and investigate it using the linear response functional formalism. The result is a careful accounting of all sound and vortex interactions close to local equilibrium. In particular, we demonstrate that the mixing of sound waves and vortices via polarization, first observed in the ideal fluid limit, extends to the shear mode once dissipative effects are included. This provides a realization within Lagrangian hydrodynamics of the symmetric shear polarization contribution recently advocated from transport and Zubarev approaches as well as phenomenological considerations. Once causal relaxational dynamics is included, this effect, seemingly puzzling because it results in a nondissipative coupling between a transient mode to an equilibrium quantity, can be understood as a competition between the Israel-Stewart and the polarization relaxation timescale, and a breakdown of local Markovianity. We close by discussing phenomenological implications of these results.

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.

Dissipative charged homologous model for cluster of stars in f(R,T) gravity

In this paper, we discuss homologous model for cluster of stars in f ( R , T ) gravity. For this purpose, we use f ( R , T ) = R + K ( − T ) n model to incorporate exotic terms in the system. The quasi-static approximations are being imposed onto the shear-free dissipative relativistic self-gravitating charged fluid. It is found that non-dissipative case can easily be reduced to linear homology law in the Newtonian regime. In dissipative scenario, this condition exhibits that the linear homology law for a fluid element is applicable only if we apply the homology conditions on temperature, emission rate and charge associated to baryonic matter. For dark matter, it depends upon the emission of gravitational dissipation. We also deduce that the shear-free and homogeneous expansion rate conditions are equivalent to the homology conditions only in the Newtonian limit. Furthermore, the deviation from homology conditions leads to thermal peeling effects. We use PSR J 1614 − 2230 data with f ( R , T ) field equations for which the graphical analysis shows that the physical variables of baryonic matter like density, pressures and dissipation are suppressed in the presence of dark matter. Thus dark matter has a significant relevance in the emergence of homologous evolution of stellar cluster.

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.

Analytic solutions of relativistic dissipative spin hydrodynamics with Bjorken expansion

We have studied analytically the longitudinally boost-invariant motion of a relativistic dissipative fluid with spin. We have derived the analytic solutions of spin density and spin chemical potential as a function of proper time $\ensuremath{\tau}$ in the presence of the viscous tensor and the second order relaxation time corrections for spin. Interestingly, analogous to the ordinary particle number density and chemical potential, we find that the spin density and spin chemical potential decay as $\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1}$ and $\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/3}$, respectively. These solutions can serve both to gain insight on the dynamics of spin polarization in relativistic heavy-ion collisions and as test beds for further numerical codes.

Covariant approach to relativistic large-eddy simulations: The fibration picture

Models of turbulent flows require the resolution of a vast range of scales, from large eddies to small-scale features directly associated with dissipation. As the required resolution is not within reach of large scale numerical simulations, standard strategies involve a smoothing of the fluid dynamics, either through time averaging or spatial filtering. These strategies raise formal issues in general relativity, where the split between space and time is observer dependent. To make progress, we develop a new covariant framework for filtering/averaging based on the fibration of spacetime associated with fluid elements and the use of Fermi coordinates to facilitate a meaningful local analysis. We derive the resolved equations of motion, demonstrating how "effective" dissipative terms arise because of the coarse-graining, and paying particular attention to the thermodynamical interpretation of the resolved quantities. Finally, as the smoothing of the fluid-dynamics inevitably leads to a closure problem, we propose a new closure scheme inspired by recent progress in the modelling of dissipative relativistic fluids, and crucially, demonstrate the linear stability of the proposed model.

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.

Linearizing a non-linear formulation for general relativistic dissipative fluids

Fully non-linear equations of motion for dissipative general relativistic multi-fluids can be obtained from an action principle involving the explicit use of lower dimensional matter spaces. More traditional strategies for incorporating dissipation—like the famous Müller–Israel–Stewart model—are based on expansions away from equilibrium defined, in part, by the laws of thermodynamics. The goal here is to build a formalism to facilitate comparison of the action-based results with those based on the traditional approach. The first step of the process is to use the action-based approach itself to construct self-consistent notions of equilibrium. Next, first-order deviations are developed directly on the matter spaces, which motivates the latter as the natural arena for the underlying thermodynamics. Finally, we identify the dissipation terms of the action-based model with first-order ‛thermodynamical’ fluxes, on which the traditional models are built. The description is developed in a general setting so that the formalism can be used to describe multi-fluid systems, for which causal and stable models are not yet available. As an illustration of the approach, a simple application of a single viscous fluid is considered and, even though the expansion is halted at first order, we sketch how a causal response can be implemented through Cattaneo-type equations.

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.

On extended thermodynamics: From classical to the relativistic regime

The recent monumental detection of gravitational waves by LIGO, the subsequent detection by the LIGO/VIRGO observatories of a binary neutron star merger seen in the gravitational wave signal [Formula: see text], the first photo of the event horizon of the supermassive black hole at the center of Andromeda galaxy released by the EHT telescope and the ongoing experiments on Relativistic Heavy Ion Collisions at the BNL and at the CERN, demonstrate that we are witnessing the second golden era of observational relativistic gravity. These new observational breakthroughs, although in the long run would influence our views regarding this Kosmos, in the short run, they suggest that relativistic dissipative fluids (or magnetofluids) and relativistic continuous media play an important role in astrophysical-and also subnuclear-scales. This realization brings into the frontiers of current research theories of irreversible thermodynamics of relativistic continuous media. Motivated by these considerations, we summarize the progress that has been made in the last few decades in the field of nonequilibrium thermodynamics of relativistic continuous media. For coherence and completeness purposes, we begin with a brief description of the balance laws for classical (Newtonian) continuous media and introduce the classical irreversible thermodynamics (CIT) and point out the role of the local-equilibrium postulate within this theory. Tangentially, we touch the program of rational thermodynamics (RT), the Clausius–Duhem inequality, the theory of constitutive relations and the emergence of the entropy principle in the description of continuous media. We discuss at some length, theories of non equilibrium thermodynamics that sprang out of a fundamental paper written by Müller in 1967, with emphasis on the principles of extended irreversible thermodynamics (EIT) and the rational extended irreversible thermodynamics (REIT). Subsequently, after a brief introduction to the equilibrium thermodynamics of relativistic fluids, we discuss the Israel–Stewart transient (or causal) thermodynamics and its main features. Moreover, we introduce the Liu–Müller–Ruggeri theory describing relativistic fluids. We analyze the structure and compare this theory to the class of dissipative relativistic fluid theories of divergent type developed in the late 1990 by Pennisi, Geroch and Lindblom. As far as theories of nonequilibrium thermodynamics of classical media are concerned, it is fair to state that substantial progress has been made and many predictions of the extended theories have been placed under experimental scrutiny. However, at the relativistic level, the situation is different. Although the efforts aiming to the development of a sensible theory (or theories) of nonequilibrium thermodynamics of relativistic fluids (or continuous media) spans less than a half-century, and even though enormous steps in the right direction have been taken, nevertheless as we shall see in this review, still a successful theory of relativistic dissipation is lacking.

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.