Microscopic model for relativistic hydrodynamics of ideal plasmas

Abstract

Relativistic hydrodynamics of classic plasmas is derived from the microscopic model in the limit of ideal plasmas, while the microscopic description assumes the level of dynamics of individual particles and the kinetic models look as the macroscopic models in this approach. The chain of equations is constructed step by step starting from the concentration evolution. It happens that the energy density and the momentum density do not appear at such approach, but new relativistic hydrodynamic variables appear in the model. These variables has no nonrelativistic analogs, but they are reduced to the concentration, the particle current, the pressure (the flux of the particle current) if relativistic effects are dropped. These variables are reduced to functions of the concentration, the particle current, the pressure if the thermal velocities are dropped in compare with the relativistic velocity field. Final equations are presented in the monopole limit of the mean-field (the self-consistent field) approximation. Hence, the contributions of the electric dipole moment, magnetic dipole moment, electric quadrupole moment, etc., of the macroscopically infinitesimal element of volume appearing in derived equations are dropped.

Graphical abstract

Appendix: Role of relativistic temperatures in definition of hydrodynamic functions

Here we present a citation from p.2 (before eq. 5) of Ref. [21] “For a relativistic system, the mass density $\rho (t, x)$ is not a good degree of freedom because it does not account for kinetic energy that may become sizable for motions close to the speed of light. Instead, it is useful to replace it by the total energy density $\epsilon (t, x)$, which reduces to $\rho$ in the nonrelativistic limit.” It is true as the relation of functions, but the mass density $\rho (t, x)$ (or, more precisely, the charge density $(q/m)\rho (t, x)$) is the source of electromagnetic field in the Maxwell’s equations. So, it cannot be eliminated from the model (see Eq.  71 and the comment after it).

Equation 6 of Ref. [21] being true for the single particle or cold fluid cannot be so easily justified for the hot fluids since the coordinate and time are independent variables in any field theory including the hydrodynamics. Therefore, values $d{{\textbf {x}}}/dt$ and $dx^{\mu }/dt$ have no physical meaning. They have no relation to the velocity field. Equations 5 and 6 of Ref. [21] give background for the Lorentz covariant theory, but it is inconsistent for the hot fluids since it contradicts the microscopic picture. Since the introduced in Ref. [21] four-velocity looks inconsistent, we have same conclusion about the energy-momentum tensor presented by equation (8) Ref. [21]. It is possible that same equations can be justified in proper way.

Theories of relativistic viscous hydrodynamics [27] like “Hydrodynamics as a gradient expansion,” “Conformal viscous hydrodynamics,” “Nonconformal hydrodynamics,” etc., are presented on the macroscopic level, so they required microscopic justification like one presented in this paper.

References [20, 22,23,24,25] follow the same phenomenological picture as Ref. [21]. As an example, on page 57 of Ref. [24] authors consider “Effective Action Formulation” instead of some microscopic picture. It is major difference from work presented here. Moreover, on page 62 of Ref. [24], the chapter 3 entitled “Microscopic Theory Background” is presented. However, notion “Microscopic” refers to the kinetic theory which is constructed in simplified macroscopic way, as it is demonstrated by equations (3.1) and (3.2). The microscopic picture is not so simple in area of high-energy physics. However, we can expect that the quantum-relativistic hydrodynamics discussed in Refs. [23,24,25] should have nonquantum relativistic limit. Strict derivation based on individual relativistic motion of classic particles is addressed here. So, problem unveiled here are expected to appear in the quantum regime. Quantum relativistic regime is properly addressed in Refs. [38,39,40, 50] (see also simplified approach in Ref. [51]). These are two limits of quantum-relativistic hydrodynamics.

Ref. [28] is entitled “Derivation of transient relativistic fluid dynamics from the Boltzmann equation.” Being focused on the microscopic justification of hydrodynamic model, we should to point out that any kinetic model also requires some microscopic justification. So, derivation from the kinetic model does not answer the formulated problem.

Let us also comment on the method of description of distributed mediums presented in Ref. [19]. Particularly, we focus on chapter II, where eq. 1 on page 22 gives the method of averaging of physical quantities. On the first step, authors used unspecified distribution function to average microscopic concentration of particles. It creates a logical gap in analysis since the evolution determined by the interaction of particles is hidden via the probabilistic approach. In particular, let us comment on the method of consideration of time derivatives of macroscopic function presented with eq. 2 on page 22 in Ref. [19]. Authors suggest that the time derivative acts on one of two functions under integral, but this mathematically incorrect step is not justified from any physical requirements. Single reason to use such method of calculation of the time derivative is an attempt to obtain well-known equations from definition given by eq. 1. In contrast with Ref. [19], we use specific method of averaging on the space volume and consider the straightforward derivation of macroscopic equations using the microscopic dynamics of individual particles.