On the nature of the so-called generic instabilities in dissipative relativistic hydrodynamics

It is shown that the so-called generic instabilities that appear in the framework of relativistic linear irreversible thermodynamics (LIT), describing the fluctuations of a simple fluid close to equilibrium, arise due to the coupling of heat with hydrodynamic acceleration which appears in Eckart’s formalism of relativistic irreversible thermodynamics. Further, we emphasize that such behavior should be interpreted as a contradiction to the postulates of LIT, namely a violation of Onsager’s hypothesis on the regression of fluctuations, and not as fluid instabilities. Such contradictions can be avoided within a relativistic linear framework if a Meixner-like approach to the phenomenological equations is employed.     

Comparing the first- and second-order theories of relativistic dissipative fluid dynamics using the 1+1 dimensional relativistic flux corrected transport algorithm

Focusing on the numerical aspects and accuracy we study a class of bulk viscosity driven expansion scenarios using the relativistic Navier–Stokes and truncated Israel–Stewart form of the equations of relativistic dissipative fluids in 1+1 dimensions. The numerical calculations of conservation and transport equations are performed using the numerical framework of flux corrected transport. We show that the results of the Israel–Stewart causal fluid dynamics are numerically much more stable and smoother than the results of the standard relativistic Navier–Stokes equations.     

Kubo formulas for relativistic fluids in strong magnetic fields

Magnetohydrodynamics of strongly magnetized relativistic fluids is derived in the ideal and dissipative cases, taking into account the breaking of spatial symmetries by a quantizing magnetic field. A complete set of transport coefficients, consistent with the Curie and Onsager principles, is derived for thermal conduction, as well as shear and bulk viscosities. It is shown that in the most general case the dissipative function contains five shear viscosities, two bulk viscosities, and three thermal conductivity coefficients. We use Zubarev’s non-equilibrium statistical operator method to relate these transport coefficients to correlation functions of the equilibrium theory. The desired relations emerge at linear order in the expansion of the non-equilibrium statistical operator with respect to the gradients of relevant statistical parameters (temperature, chemical potential, and velocity.) The transport coefficients are cast in a form that can be conveniently computed using equilibrium (imaginary-time) infrared Green’s functions defined with respect to the equilibrium statistical operator.     

Relativistic non-instantaneous action-at-a-distance interactions

Relativistic action-at-a-distance theories with interactions that propagate at the speed of light in vacuum are investigated. We consider the most general action depending on the velocities and relative positions of the particles. The Poincaré invariant parameters that label successive events along the world lines can be identified with the proper times of the particles provided that certain conditions are imposed on the interaction terms in the action. Further conditions on the interaction terms arise from the requirement that mass be a scalar. A generic class of theories with interactions that satisfy these conditions is found. The relativistic equations of motion for these theories are presented. We obtain exact circular orbits solutions of the relativistic one-body problem. The exact relativistic one-body Hamiltonian is also derived. The theory has three components: a linearly rising potential, a Coulomb-like interaction and a dynamical component to the Poincaré invariant mass. At the quantum level we obtain the generalized Klein–Gordon–Fock equation and the Dirac equation.     

Relativistic two-stream instability

We study the (local) propagation of plane waves in a relativistic, non- dissipative, two-fluid system, allowing for a relative velocity in the “background” configuration. The main aim is to analyze relativistic two-stream instability. This instability requires a relative flow—either across an interface or when two or more fluids interpenetrate—and can be triggered, for example, when one-dimensional plane-waves appear to be left-moving with respect to one fluid, but right-moving with respect to another. The dispersion relation of the two-fluid system is studied for different two-fluid equations of state: (i) the “free” (where there is no direct coupling between the fluid densities), (ii) coupled, and (iii) entrained (where the fluid momenta are linear combinations of the velocities) cases are considered in a frame-independent fashion (e.g.no restriction to the rest-frame of either fluid). As a by-product of our analysis we determine the necessary conditions for a two-fluid system to be causal and absolutely stable and establish a new constraint on the entrainment.     

Generalized entropy and transport coefficients of hadronic matter

We use the generalized entropy four-current of the Müller-Israel-Stewart (MIS) theories of relativistic dissipative fluids to obtain information about fluctuations around equilibrium states. This allows one to compute the non-classical coefficients of the entropy 4-flux in terms of the equilibrium distribution functions. The Green-Kubo formulae are used to compute the standard transport coefficients from the fluctuations of entropy due to dissipative fluxes.     

Quantum information theory

A conceptual analysis of the classical information theory of Shannon (1948) shows that this theory cannot be directly generalized to the usual quantum case. The reason is that in the usual quantum mechanics of closed systems there is no general concept of joint and conditional probability. Using, however, the generalized quantum mechanics of open systems (A. Kossakowski 1972) and the generalized concept of observable (“semiobservable”, E.B. Davies and J.T. Lewis 1970) it is possible to construct a quantum information theory being then a straightforward generalization of Shannon's theory.     

From kinetic theory to dissipative fluid dynamics

We present the results of deriving the Israel-Stewart equations of relativistic dissipative fluid dynamics from kinetic theory via Grad’s 14-moment expansion. Working consistently to second order in the Knudsen number, these equations contain several new terms which are absent in previous treatments.     

Shock wave formation in hot nuclear matter

Assuming that nuclear matter can be treated as a perfect fluid, we study the propagation of perturbations in the baryon density at high temperature. The equation of state is derived from the non-linear Walecka model. The expansion of the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations lead to the breaking wave equation for the density perturbation. We solve it numerically for this perturbation and follow the propagation of the initial pulses.     

Geometric gradient-flow dynamics with singular solutions

The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy’s Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.     

The kinetic theory of the perturbations in the isotropic universe

The short wave and long wave gravitational perturbations in the isotropic universe are considered by using the general relativistic kinetic theory. It is shown that the short wave perturbations propagate in the ultrarelativistic collisionless gas with the velocity of light.     

On relativistic thermodynamics

The thermodynamics of moving bodies is developed from first principles. To do this, it is necessary to augment the laws of thermodynamics with a new principle, which asserts the impossibility of thermal equilibrium between bodies in relative motion. Clausius' theorem is generalized to heat flow between moving systems, and leads naturally to the identification of heat and temperature as Lorentz scalars. The formulation of relativistic statistical mechanics is carried out and the correspondence with classical quantities is made. The quantum distribution laws are generalized to the relativistic case, and are found to differ from their accepted relativistic form.     

Relativistic hydrodynamics – causality and stability

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 choice of the velocity field is applied.     

Eikonal Approximation to 5D Wave Equations and the 4D Space-Time Metric

We apply a method analogous to the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold, using a method which identifies the symplectic structure of the corresponding mechanics, to the five dimensional generalization of Maxwell theory required by the gauge invariance of Stueckelberg's covariant classical and quantum dynamics. In this way, we demonstrate, in the eikonal approximation, the existence of geodesic motion for the flow of mass in a four dimensional pseudo-Riemannian manifold. These results provide a foundation for the geometrical optics of the five dimensional radiation theory and establish a model in which there is mass flow along geodesics. We then discuss the interesting case of relativistic quantum theory in an anisotropic medium as well. In this case the eikonal approximation to the relativistic quantum mechanical current coincides with the geodesic flow governed by the pseudo-Riemannian metric obtained from the eikonal approximation to solutions of the Stueckelberg–Schrödinger equation. The locally symplectic structure which emerges is that of a generally covariant form of Stueckelberg's mechanics on this manifold.     

Effects of electromagnetic field on energy density inhomogeneity in self-gravitating fluids

This paper is devoted to study the effects of electromagnetic field on the energy density inhomogeneity in the relativistic self-gravitating fluids for spherically symmetric spacetime. Two important equations of the Weyl tensor are formulated which help to analyze the energy density inhomogeneity in this scenario. We investigate two types of fluids, i.e., non-dissipative and dissipative. The non-dissipative fluid further includes dust, locally isotropic, and locally anisotropic charged fluids. We explore the effects of different factors on energy density inhomogeneity in all these cases, in particular, the effect of charge.     

Stable first-order particle-frame relativistic hydrodynamics for dissipative systems

We propose a stable first-order relativistic dissipative hydrodynamic equation in the particle frame (Eckart frame) for the first time. The equation to be proposed was in fact previously derived by the authors and a collaborator from the relativistic Boltzmann equation. We demonstrate that the equilibrium state is stable with respect to the time evolution described by our hydrodynamic equation in the particle frame. Our equation may be a proper starting point for constructing second-order causal relativistic hydrodynamics, to replace Eckart's particle-flow theory.     

Dissipative forces and the algebra of operators in stochastic quantum mechanics

Davidson's construction of a Hilbert space and of quantum operators on the basis of the Fényes-Nelson stochastic mechanics is extended to the case in which a dissipative force linear in the velocity is present. The hamiltonian becomes a nonlinear operator but the position and linear momentum operators are the same as in ordinary quantum mechanics.     

A comment on the stochastic quantization of friction

It is shown that Hashimoto's recent construction of a nonlinear wave equation on the basis of stochastic mechanics for a dissipative system submitted to a general velocity-dependent frictional force is not valid, except for the trivial one-dimensional case.     

Relativistic effects of laser doppler velocimeters using one light beam with two frequencies

Relativistic effects of laser Doppler velocimeters (LDV's) are discussed and novel LDV systems are proposed. If the direction of the scattered light makes a right angle with the flow direction, relativistic effects completely disappear no matter how high the velocity of a moving particle becomes. The proposed LDV's involve that the velocity can be measured from one scattered light beam with two different single frequencies. It is also predicted that the usual optical heterodyne-detection techniques can be made applicable to measure even ultra-high velocities up to the region where relativistic effects should be taken into account.