Relativistic Hydrodynamics Research Articles

Relativistic hydrodynamics from the single-generator bracket formalism of nonequilibrium thermodynamics

Abstract We employ the generalized bracket formalism of nonequilibrium thermodynamics by Beris and Edwards to derive Lorentz-covariant time-evolution equations for an imperfect fluid with viscosity, dilatational viscosity, and thermal conductivity. Following closely the analysis presented by Öttinger (Physica A, 259, 1998, 24–42; Physica A, 254, 1998, 433–450) to the same problem but for the GENERIC formalism, we include in the set of hydrodynamic variables a covariant vector playing the role of a generalized thermal force and a covariant tensor closely related to the velocity gradient tensor. In our work here, we derive first the nonrelativistic equations and then we proceed to obtain the relativistic ones by elevating the thermal variable to a four-vector, the mechanical force variable to a four-by-four tensor, and by representing the Hamiltonian of the system with the time component of the energy-momentum tensor. For the Poisson and dissipation brackets we assume the same general structure as in the nonrelativistic case, but with the phenomenological coefficients in the dissipation bracket describing friction to heat and viscous effects being properly constrained for the resulting dynamic equations to be manifest Lorentz-covariant. The final relativistic equations are identical to those derived by Öttinger but the present approach seems to be more general in the sense that one could think of alternative forms of the phenomenological coefficients describing friction that could ensure Lorentz-covariance.

Provably convergent Newton–Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics

The relativistic hydrodynamics (RHD) equations have three crucial intrinsic physical constraints on the primitive variables: positivity of pressure and density, and subluminal fluid velocity. However, numerical simulations can violate these constraints, leading to nonphysical results or even simulation failure. Designing genuinely physical-constraint-preserving (PCP) schemes is very difficult, as the primitive variables cannot be explicitly reformulated using conservative variables due to relativistic effects. In this paper, we propose three efficient Newton–Raphson (NR) methods for robustly recovering primitive variables from conservative variables. Importantly, we rigorously prove that these NR methods are always convergent and PCP, meaning they preserve the physical constraints throughout the NR iterations. The discovery of these robust NR methods and their PCP convergence analyses are highly nontrivial and technical. Our NR methods are versatile and can be seamlessly incorporated into any RHD schemes that require the recovery of primitive variables. As an application, we apply them to design PCP finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the RHD equations. Our PCP HWENO schemes incorporate high-order HWENO reconstruction, a PCP limiter, and strong-stability-preserving time discretization. We rigorously prove the PCP property of the fully discrete schemes using convex decomposition techniques. Moreover, we suggest the characteristic decomposition with rescaled eigenvectors and scale-invariant nonlinear weights to enhance the performance of the HWENO schemes in simulating large-scale RHD problems. Several demanding numerical tests are conducted to demonstrate the robustness, accuracy, and high resolution of the proposed PCP HWENO schemes and to validate the efficiency of our NR methods.

Exact Riemann solver for non-convex relativistic hydrodynamics

We present the general analytical solution of the Riemann problem (decay of a jump discontinuity) for non-convex relativistic hydrodynamics. In convex dynamics, an elementary nonlinear wave, i.e. a rarefaction or a shock, originates at the discontinuity and travels towards one of the initial states. Between the left and right waves, an equilibrium state appears represented by a contact discontinuity. The exact solution to the Riemann problem in convex relativistic hydrodynamics was first addressed by Martí & Müller (J. Fluid Mech., vol. 258, 1994, pp. 317–333). In non-convex dynamics, two sequences of elementary nonlinear waves move towards the left and right initial states. Solving the Riemann problem involves determining the types of wave developing and the equilibrium state where they coincide. The procedure consists of constructing the wave curves associated with the nonlinear waves in the pressure–velocity phase space, where the intersection of the wave curves indicates the equilibrium state. We describe the relation between the wave curves, the explicit formulas for their calculation, and the outline of the process for a correct derivation and representation of the waves in the spatial domain. We present examples of the exact solution of a Riemann problem that illustrate the complex phenomena of non-convex dynamics by using the phenomenological non-convex equation of state proposed by Ibáñez et al. (Mon. Not. R. Astron. Soc., vol. 476, 2017, pp. 1100–1110).

Linearly stable and causal relativistic first-order spin hydrodynamics

Linearly stable and causal relativistic first-order spin hydrodynamics

Impact of vorticity and viscosity on the hydrodynamic evolution of hot QCD medium

The strongly interacting transient state of quark-gluon plasma (QGP) medium created in ultra-relativistic collisions survives for a duration of a few fm/c. The spacetime evolution of QGP crucially depends on the equation of state (EoS), vorticity, viscosity, and external magnetic field. In the present study, we obtain the lifetime of a vortical QGP fluid within the ambit of relativistic second-order viscous hydrodynamics. We observe that the coupling of vorticity and viscosity significantly increases the lifetime of vortical QGP. The inclusion of a static magnetic field, vorticity, and viscosity makes the evolution slower. However, the static magnetic field slightly decreases the QGP lifetime by accelerating the evolution process for a non-rotating medium. We also report the rate of change of vorticity in the QGP, which will be helpful in studying the behavior of the medium in detail.

The quantum exchange effects on the electromagnetic solitonic excitations in warm plasma

The phenomena related to the high energy regime in a plasma medium are studied in the framework of relativistic quantum hydrodynamics at the classical level. This theory leads to nonlinear equations even at the weak relativistic regime. These nonlinearities give rise to interesting and novel phenomena in the dynamics of the medium. Here, we consider these effects on the dynamics of a warm plasma environment. In addition, we study the effects of the exchange interaction between electrons on the dynamic of such a plasma. This interaction is a quantum mechanical effect and its strength is influenced by the spin polarization ratio of the electrons. Both the analytical and the numerical solutions are presented which are stable and bounded soliton waves. For the analytical solutions, an extended tanh approach, and for the numeric approach, the techniques of solving eigenvalue problems are applied. The behavior of the vector potential and the density under the spin polarization effects and thermal effects in terms of the speed of the wave are discussed.

High-resolution simulations of LS 5039

Context. We present an analysis of our high-resolution relativistic hydrodynamics model of the stellar- and pulsar-wind interactions in the LS-5039 system. Aims. With our high-resolution simulation covering three orbital periods, we analyse the impact of turbulence with a particular focus on short-term and orbit-to-orbit variations. Methods. Our model uses a relativistic hydrodynamical description of the wind interaction in the LS-5039 system, assuming a pulsar wind-driven scenario. The corresponding system of equations was solved using the finite-volume code CRONOS. We computed the statistical quantities, which are also relevant for particle acceleration in this system, from the results of multiple consecutive timesteps. Results. In our simulation, we find that the previously observed shock structures related to the wind-collision region (WCR), including the pulsar-wind termination, are dynamically influenced by orbital motion. From our high-resolution simulation, we find high turbulence levels following from instabilities driven at the WCR. These instabilities lead to strong fluctuations of several dynamical quantities, especially around and after apastron. These fluctuations are expected to impact the particle transport and especially the related emission of non-thermal radiation. As an important example, the region for which gamma-ray emission has been found to be boosted due to relativistic beaming (in previous studies) shows strong variations in size both on short and orbital timescales. Conclusions. Using a large computational domain together with a high spatial resolution allowed us to carry out a detailed study of fluctuations in stellar- and pulsar-wind interactions. The results indicate a possible influence on the non-thermal emission from this system, which will be analysed with dedicated simulations in a forthcoming publication.

Toward Powerful Probes of Neutrino Self-Interactions in Supernovae.

Neutrinos remain mysterious. As an example, enhanced self-interactions (νSI), which would have broad implications, are allowed. At the high neutrino densities within core-collapse supernovae, νSI should be important, but robust observables have been lacking. We show that νSI make neutrinos form a tightly coupled fluid that expands under relativistic hydrodynamics. The outflow becomes either a burst or a steady-state wind; which occurs here is uncertain. Though the diffusive environment where neutrinos are produced may make a wind more likely, further work is needed to determine when each case is realized. In the burst-outflow case, νSI increase the duration of the neutrino signal, and even a simple analysis of SN 1987A data has powerful sensitivity. For the wind-outflow case, we outline several promising ideas that may lead to new observables. Combined, these results are important steps toward solving the 35-year-old puzzle of how νSI affect supernovae.

Collective dynamics of polarized spin-half fermions in relativistic heavy-ion collisions

Standard relativistic hydrodynamics has been successful in describing the properties of the strongly interacting matter produced in the heavy-ion collision experiments. Recently, there has been a significant theoretical advancement in this field to explain spin polarization of hadrons emitted in these processes. Although current models have successfully explained some of the experimental data based on the coupling between spin polarization and vorticity of the medium, they still lack a clear understanding of the differential measurements. This is commonly interpreted as an indication that the spin needs to be treated as an independent degree of freedom whose dynamics is not entirely bound to flow circulation. In particular, if the spin is a macroscopic property of the system, in equilibrium its dynamics should follow hydrodynamic laws. Here, we develop a framework of relativistic hydrodynamics which includes spin degrees of freedom from the quantum kinetic theory for Dirac fermions and use it for modeling the dynamics of matter. Following experimental observations, we assume that the polarization effects are small and derive conservation laws for the net baryon current, the energy–momentum tensor and the spin tensor based on the de Groot–van Leeuwen–van Weert definitions of these currents. We present various properties of the spin polarization tensor and its components, analyze the propagation properties of the spin polarization components, and derive the spin-wave velocity for arbitrary statistics. We find that only the transverse spin components propagate, analogously to the electromagnetic waves. Finally, using our framework, we study the space–time evolution of the spin polarization for the systems respecting certain space–time symmetries and calculate the mean spin polarization per particle, which can be compared to the experimental data. We find that, for some observables, our spin polarization results agree qualitatively with the experimental findings and other model calculations.

Relativistic second-order spin hydrodynamics: An entropy-current analysis

Relativistic second-order spin hydrodynamics: An entropy-current analysis

Suppression of spin polarization as an indicator of QCD critical point

We study the impact of the QCD critical point (CP) on the spin polarization of Lambda -hyperon generated by the thermal vorticity in viscous quark gluon plasma (QGP). The equations of the relativistic causal viscous hydrodynamics have been solved numerically in (3+1) dimensions to evaluate the thermal vorticity. The effects of the CP have been incorporated through the equation of state (EoS) and the scaling behavior of the transport coefficients. We observe a significant suppression of the spin polarization around mid-rapidity region induced by the CP and propose that such changed behaviour of the spin polarization can be used to detect the CP. To the best of our knowledge this is the first study on the effects of critical point on the spin polarization in QCD matter.

Microscopic model for relativistic hydrodynamics of ideal plasmas

Microscopic model for relativistic hydrodynamics of ideal plasmas

Transport coefficients in AdS/CFT and quantum gravity corrections due to a functional measure

The presence of a functional measure is scrutinized on both sides of the dual gauge/gravity correspondence. Corrections to the transport coefficients in relativistic hydrodynamics are obtained using the linear response procedure. In particular, using first-order hydrodynamics, the shear viscosity, entropy density, diffusion constant, and speed of sound are shown not to acquire any corrections from the functional measure of gravity, for a Minkowski background metric. On the other hand, the energy density, the pressure, the relaxation time, the bulk viscosity, the decay rate of sound waves, and coefficients of conformal traceless tensor fields, are shown to carry significant quantum corrections due to the functional measure, even for a flat background. They all acquire an imaginary part that reflects the instability of the strongly-coupled fluids on the boundary CFT. This opens up the possibility of testing quantum gravity with the quark-gluon plasma.

Pseudogauge freedom and the SO(3) algebra of spin operators

The energy-momentum and spin tensors for a given theory can be replaced by alternative expressions that obey the same conservation laws for the energy, linear momentum, as well as angular momentum but, however, differ by the local redistribution of such quantities (with global energy, linear momentum, and angular momentum remaining unchanged). This arbitrariness is described in recent literature as the pseudogauge freedom or symmetry. In this letter, we analyze several pseudogauges used to formulate the relativistic hydrodynamics of particles with spin 12 and conclude that the canonical version of the spin tensor has an advantage over other forms as only the canonical definition defines the spin operators that fulfill the SO(3) algebra of angular momentum. This result sheds new light on the results encountered in recent papers demonstrating pseudogauge dependence of various physical quantities. It indicates that for spin-polarization observables, the canonical version is fundamentally better suited for building a connection between theory and experiment.

Equations of Quantum Relativistic Hydrodynamics and Soliton Solutions in Describing Nucleus–Nucleus Collisions

Equations of Quantum Relativistic Hydrodynamics and Soliton Solutions in Describing Nucleus–Nucleus Collisions

Bjorken Flow Revisited: Analytic and Numerical Solutions in Flat Space-Time Coordinates

In this work we provide analytic and numerical solutions for the Bjorken flow, a standard benchmark in relativistic hydrodynamics providing a simple model for the bulk evolution of matter created in collisions between heavy nuclei. We consider relativistic gases of both massive and massless particles, working in a ( 2 + 1 ) and ( 3 + 1 ) Minkowski space-time coordinate system. The numerical results from a recently developed lattice kinetic scheme show excellent agreement with the analytic solutions.

Three‐dimensional stationary supersonic flows with axial symmetry in relativistic hydrodynamics

AbstractThe study of supersonic flows in relativistic hydrodynamics (RHDs) is of some interest for cosmology. This paper studies two types of three‐dimensional (3D) stationary supersonic flows with axial symmetry in RHDs: supersonic flows expanding in vacuum and supersonic flows against an infinite curved cone. To study supersonic flows expanding in vacuum, we consider a Cauchy problem for the 3D steady RHD equations with initial data that are a combination of an axisymmetric supersonic flow in a circular domain and vacuum in the remaining domain. Global existence of a piecewise smooth solution expanding in vacuum to the Cauchy problem is obtained, provided that the initial mass‐energy density is sufficiently small. When a supersonic flow arrives and hits a curved cone, a shock wave will arise around the cone. Moreover, if the vertex angle of the cone is less than a critical angle, then the shock will be attached to the vertex of the cone. We obtain the global existence of attached conical shock waves, provided that the curved cone satisfies some suitable conditions and the speed of the incoming flow is close to a limit speed.

High-order finite-difference entropy stable schemes for two-fluid relativistic plasma flow equations

In this article, we propose high-order finite-difference entropy stable schemes for the two-fluid relativistic plasma flow equations. This is achieved by exploiting the structure of the equations, which consists of three independent flux components. The first two components describe the ion and electron flows, which are modeled using the relativistic hydrodynamics equation and the third component is Maxwell's equations. The coupling of the ion and electron flows and electromagnetic fields is via source terms only, but the source terms do not affect the entropy evolution. To design semi-discrete entropy stable schemes, we extend the entropy stable schemes for relativistic hydrodynamics in [1] to three dimensions. This is then coupled with entropy stable discretization of the Maxwell's equations. Finally, we use SSP-RK schemes to discretize in time. We also propose ARK-IMEX schemes to treat the stiff source terms; the resulting nonlinear set of algebraic equations is local (at each discretization point) and hence can be solved cheaply using the Newton's Method. The proposed schemes are then tested using various test problems to demonstrate their stability, accuracy and efficiency.

Evolution of Nonlinear Wave Pulses in the sine-Gordon Equation Theory

We give the solution of the Whitham modulation equations for envelopes of pulses evolving according to the sine-Gordon equation. The Whitham equations are interpreted as the equations of relativistic hydrodynamics and their solving is reduced by the hodograph method to solving a linear partial differential equation. We describe the class of solutions of this equation with separation of variables and illustrate the theory by an example of the nonlinear wave packet evolution accompanied by its shrinking and decrease of the number of oscillations in the Whitham nonlinear region.

Effects of the QCD critical point on the spectra and flow coefficients of hadrons

The space-time evolution of the hot and dense fireball of quarks and gluons produced in ultrarelativistic heavy-ion collisions at nonzero baryonic chemical potential and temperature has been studied by using relativistic dissipative hydrodynamics. For this purpose, a numerical code has been developed to solve the relativistic viscous causal hydrodynamics in ($3+1$) dimensions with the inclusion of QCD critical point ($CP$) through the equation of state and scaling behavior of transport coefficients. We compute the transverse momentum spectra, directed and elliptic flow coefficients of pions and protons to comprehend the effect of $CP$ on these observables. It is found that the integration over the entire space-time history of the fireball largely obliterates the effects of $CP$ on the spectra and flow coefficients for the event averaged initial conditions considered here.