We study the causality and stability of relativistic hydrodynamics with the inclusion of the spin degree of freedom as a hydrodynamic field. We consider two specific models of spin-hydrodynamics for this purpose. A linear mode analysis for static background shows that a first-order dissipative spin-hydrodynamics remains acausal and admits instabilities. Besides, it is found that the inclusion of the spin field in hydrodynamics leads to new kinds of linear modes in the system. These new modes also exhibit instability and acausal behavior. The second model of the spin-hydrodynamics that we have considered here is equivalent to a particular second-order conventional hydrodynamics with no dissipative effects. For a static background, it is found that the linear modes of this model support the sound waves only. However, when the background has constant vorticity, then the model admits instability and acausality in certain situations. It is found that the spin-dynamics have an effect on the hydrodynamic response of the fluid. These findings point toward the need for a causal and stable theory with spin as a hydrodynamic field to describe the spin-polarized fluid.
The stability and causality of the Landau-Lifshitz theory and the Israel-Stewart type causal dissipative hydrodynamics are discussed. We show that the problem of acausality and instability are correlated in relativistic dissipative hydrodynamics and instability is induced by acausality. We further discuss the stability of the scaling solution. The scaling solution of the causal dissipative hydrodynamics can be unstable against inhomogeneous perturbations.
We analyze the breakdown of causality for the perfect fluid limit in a medium with polarizeability. We show that to restore causality a relaxation term linking vorticity and polarization, analogous to the Israel-Stewart term linking viscous forces and gradients,is required. This term provides a minimum amount of dissipation a locally thermalized relativistic medium with polarizeability must have, independently of its underlying degrees of freedom. For ferromagnetic materials an infrared acausal mode remains, which we interpret as a Banks-Casher mode signaling spontaneous magnetization. With th…
It is shown that the relativistic zero-range potential scattering surpasses Wigner's causality bound, while being consistent with causality. The relativistic theory shows in addition a richer analytic structure, such as a $K$-matrix pole necessarily accompanying the bound-state solution. Implications of these results for the effective-field theory of nuclear forces are briefly considered.
Causality and stability in relativistic dissipative hydrodynamics are important conceptual issues. We argue that causality is not restricted to hyperbolic set of differential equations. E.g. heat conduction equation can be causal considering the physical validity of the theory. Furthermore we propose a new concept of relativistic internal energy that clearly separates the dissipative and non-dissipative effects. We prove that with this choice we remove all known instabilities of the linear response approximation of viscous and heat conducting relativistic fluids. In this paper the Eckart choic…
We investigate the causality and stability of relativistic dissipative fluid dynamics in the absence of conserved charges. We perform a linear stability analysis in the rest frame of the fluid and find that the equations of relativistic dissipative fluid dynamics are always stable. We then perform a linear stability analysis in a Lorentz-boosted frame. Provided that the ratio of the relaxation time for the shear stress tensor, $τ_π$, to the sound attenuation length, $Γ_s = 4η/3(\varepsilon+P)$, fulfills a certain asymptotic causality condition, the equations of motion give rise to stable solut…
In this work, the causality and stability of a first-order relativistic dissipative hydrodynamic theory, that redefines the hydrodynamic fields from a first principle microscopic estimation, have been analyzed. A generic approach of gradient expansion for solving the relativistic transport equation has been adopted using the Chapman-Enskog iterative method. Next, the momentum dependent relaxation time approximation (MDRTA) has been employed to quantify the collision term for analytical estimation of the field correction coefficients from kinetic theory. At linear regime, in local rest frame th…
Causality and Stability of First-Order Relativistic Spin Hydrodynamics with Conserved Charges
College of Mathematics and Physics, China Three Gorges University（三峡大学数学与物理学院）；Center for Astronomy and Space Sciences and Institute of Modern Physics, China Three Gorges University（三峡大学天文与空间科学中心及现代物理研究所）；Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University（华中师范大学夸克与轻子物理教育部重点实验室及粒子物理研究所）
Effective theory arguments are used to derive the most general energy-momentum tensor of a relativistic viscous fluid with an arbitrary equation of state (in the absence of other conserved currents) that is first-order in the derivatives of the energy density and flow velocity and does not include extended variables such as in Mueller-Israel-Stewart-like theories. This energy-momentum tensor leads to a causal theory, provided one abandons the usual conventions for the out-of-equilibrium hydrodynamic variables put forward by Landau-Lifshitz and Eckart. In particular, causality requires nonzero …
Relativistic Dispersion Spectra across Lorentz boosted frames: Spurious modes and the enigma of causality
University of Edinburgh（爱丁堡大学）；National Institute of Science Education and Research（国家科学教育与研究所）；Indian Institute of Technology, Gandhinagar（甘地纳加尔印度理工学院）；West University of Timisoara（蒂米什瓦拉西大学）
Relativistic dissipative hydrodynamics including hydrodynamic fluctuations is formulated by putting an emphasis on non-linearity and causality. As a consequence of causality, dissipative currents become dynamical variables and noises appeared in an integral form of constitutive equations should be colored ones from fluctuation-dissipation relations. Nevertheless noises turn out to be white ones in its differential form when noises are assumed to be Gaussian. The obtained ifferential equations are very useful in numerical implementation of relativistic fluctuating hydrodynamics.
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 PRC 88, 021903, 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 b…
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-Molnár-Rischke (DNMR) theory significantly shrinks the casual regions compared to those in the Israel-Stewart hydrodynamics. For Au+Au collis…
Necessary conditions for causality from linearized stability at ultra-high boosts
Indian Institute of Technology, Gandhinagar（印度理工学院甘地纳加尔分校）；National Institute of Science Education and Research（国家科学教育与研究中心）；West University of Timisoara（蒂米什瓦拉西大学）
The second order Israel-Stewart-M$\ddot{u}$ller relativistic hydrodynamics has been applied to study the effects of causality on the acoustic oscillation in relativistic fluid. Causal dispersion relations have been derived with non-vanishing shear viscosity, bulk viscosity and thermal conductivity at non-zero temperature and chemical potential. These relations have been used to investigate the fluidity of Quark Gluon Plasma (QGP) at finite temperature ($T$). Results of the first order dissipative hydrodynamics have been obtained as limiting case of the second order theory. The effects of the c…
Dissipative relativistic fluid dynamics is not always causal and can favor superluminal 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 evolutions for a radial expansion are hyperbolic and how the properties of the solutions are encoded in the associated characteristic curves. The expansion dynamics is causal in relativistic sense if the characteristic velocities are smaller than th…
We present an exact solution of the relativistic Boltzmann equation for a system undergoing boost-invariant longitudinal and azimuthally symmetric transverse flow ("Gubser flow"). The resulting exact non-equilibrium dynamics is compared to 1st- and 2nd-order relativistic hydrodynamic approximations for various shear viscosity to entropy density ratios. This novel solution can be used to test the validity and accuracy of different hydrodynamic approximations in conditions similar to those generated in relativistic heavy-ion collisions.
Although the exact Bethe-Salpeter equation is certainly the appropriate field-theoretic framework to describe the non-perturbative problem of scattering and bound states, the inevitable truncations introduce inconsistencies such as loss of symmetries or incorrect one-body limit. I conjecture that these problem can be overcome if the truncation preserves the field-redefinition invariance of the exact equation. A sum rule for light-by-light scattering can provide a testing ground of this conjecture.
In this paper, I study the conditions imposed on a normal charged fluid so that the causality and stability criteria hold for this fluid. I adopt the newly developed General Frame (GF) notion in the relativistic hydrodynamics framework which states that hydrodynamic frames have to be fixed after applying the stability and causality conditions. To my purpose, I take a charged conformal matter in the flat and $3+1$ dimension to analyze better these conditions. The causality condition is applied by looking to the asymptotic velocity of sound hydro modes at the large wave number limit and stabilit…
We investigate the causality and the stability of the relativistic viscous magneto-hydrodynamics in the framework of the Israel-Stewart (IS) second-order theory, and also within a modified IS theory which incorporates the effect of magnetic fields in the relaxation equations of the viscous stress. We compute the dispersion relation by perturbing the fluid variables around their equilibrium values. In the ideal magnetohydrodynamics limit, the linear dispersion relation yields the well-known propagating modes: the Alfvén and the magneto-sonic modes.In the presence of bulk viscous pressure, the c…
A new approach is described to help improve the foundations of relativistic viscous fluid dynamics and its coupling to general relativity. Focusing on neutral conformal fluids constructed solely in terms of hydrodynamic variables, we derive the most general viscous energy-momentum tensor yielding equations of motion of second order in the derivatives, which is shown to provide a novel type of generalization of the relativistic Navier-Stokes equations for which causality holds. We show how this energy-momentum tensor may be derived from conformal kinetic theory. We rigorously prove existence, u…
We propose an effective action for first order relativistic dissipative hydrodynamics that can be used to evaluate n-point symmetrized correlation functions, taking into account thermal fluctuations of the hydrodynamic variables.
It was recently shown that the dispersion relations describing singularities of retarded two-point functions in causal quantum field theories always satisfy the fundamental inequality $\mathfrak{Im} \, ω\leq |\mathfrak{Im} \, k|$, and that several rigorous bounds on transport coefficients follow directly from such inequality. Here, we prove that the same inequality, $\mathfrak{Im} \, ω\leq |\mathfrak{Im} \, k|$, is a necessary condition for a fluid theory to be covariantly stable (i.e. stable in all frames of reference). Hence, the same bounds on transport that follow from causality in quantum…
We demonstrate that nonextensive perfect relativistic hydrodynamics ($q$-hydrodynamics) can serve as a model of the usual relativistic dissipative hydrodynamics ($d$-hydrodynamics) facilitating therefore considerably its applications. As illustration we show how using $q$-hydrodynamics one gets the $q$-dependent expressions for the dissipative entropy current and the corresponding ratios of the bulk and shear viscosities to entropy density, $ζ/s$ and $η/s$.
We use the Crooks fluctuation theorem together with Zubarev hydrodynamics to develop a bottom-up theory of hydrodynamic fluctuations. We also use thermodynamic uncertainity relations to estimate bottom-up limits to dissipative transport coefficients.
In this article, we briefly review recent progress on hydrodynamic modeling and its implementations to relativistic heavy-ion collisions at RHIC and the LHC. The related topics include: 1) initial state fluctuations, final state correlations and event-by-event hydrodynamics, 2) extracting the QGP shear viscosity from flow data, 3) flow and hydrodynamics in 5.02 A TeV p+Pb collisions.
Kinetic and hydrodynamic models describing early stages of relativistic heavy-ion collisions are discussed. We emphasise the role of the shear-bulk coupling for the correct determination of the time dependence of the bulk viscous pressure.
We argue that there is profound correspondence (the nonextensive/dissipative correspondence - NexDC) between the perfect nonextensive hydrodynamics and the usual dissipative hydrodynamics which leads to simple expression for dissipative entropy current.
We review recent progress in relativistic hydrodynamics, discussing causal and stable first-order hydrodynamics, known as BDNK theories, hydrodynamic attractors, as well as hydrodynamics near the chiral critical point and spin hydrodynamics.
We present a variational approach for relativistic ideal hydrodynamics interacting with electromagnetic fields. The momentum of fluid is introduced as the canonical conjugate variable of the position of a fluid element, which coincides with the conserved quantity derived from the Noether theorem. We further show that our formulation can reproduce the usual electromagnetic hydrodynamics which is obtained so as to satisfy the conservation of the inertia of fluid motion.
We generalize a previously known class of exact analytic solutions of relativistic perfect fluid hydrodynamics for the first time to arbitrary temperature-dependent Equation of State. We investigate special cases of this class of solutions, in particular, we present hydrodynamical solutions with the Equation of State determined from lattice QCD calculations. We discuss the phenomenological relevance of these solutions as well.
We present a first calculation of the dilepton yield and elliptic flow done with 3+1D viscous hydrodynamical simulations of relativistic heavy ion collisions at the top RHIC energy. A comparison with recent experimental data from the STAR collaboration is made.
Nonlinear Causality and Strong Hyperbolicity of Einstein-Israel-Stewart Theories of Transient Relativistic Fluid Dynamics
Department of Physics and Astronomy, University of Florence（佛罗伦萨大学物理与天文学系）；Department of Mathematics, Vanderbilt University（范德堡大学数学系）；Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte（北里奥格兰德联邦大学科学与技术学院）
