The relativistic fluid is a highly successful model used to describe the dynamics of many-particle, relativistic systems. It takes as input basic physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process, an understanding of bulk features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as “small” as heavy ions in collisions, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multiple) fluid model. We focus on the variational principle approach championed by Brandon Carter and his collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book derivation of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory.

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Relativistic fluid dynamics: physics for many different scales

Der vorliegende Band enthált sechs Beitrage, in denen punktuell und exemplarisch der Stand der Forschung beleuchtet wird. Ein Beitrag von Codling und Frasinski beschreibt in erster Linie Experimente zum noch wenig verstandenen Phânomen der dissoziativen Ionisation von Molekülen, bei dem diese in einem intensiven Laserfeld mehrere Elektronén verlieren und in Ionen mit hoher kinetischer Energie zerfallen. Ein wesentlicher Teil dieses Beitrags widmet sich einer von den Autoren entwickelten Auswertungstechnik, welche die Korrelationen zwischen den Fiugzeiten emittierter Ionen und Elektronén sichtbar macht. Es folgen drei Beitrage zur Théorie. Schmelcher und Cederbaum beschreiben Effekte starker statischer Felder, die nur dann zu verstehen sind, wenn die Nicht-Separation der Schwerpunktsbewegung im auBeren Magnetfeld korrekt beriicksichtigt wird. Dies fiihrt

Extended Irreversible Thermodynamics

We review the progress made in extended irreversible thermodynamics during the ten years that have elapsed since the publication of our first review on the same subject (Rep. Frog. Phys. 1988 51 1105-72). During this decade much effort has been devoted to achieving a better understanding of the fundamentals and a broadening of the domain of applications. The macroscopic formulation of extended irreversible thermodynamics is reviewed and compared with other non-equilibrium thermodynamic theories. The foundations of EIT are discussed on the bases of information theory, kinetic theory, stochastic phenomena and computer simulations. Several significant applications are presented, some of them of considerable practical interest (non-classical heat transport, polymer solutions, non-Fickian diffusion, microelectronic devices, dielectric relaxation), and some others of special theoretical appeal (superfluids, nuclear collisions, cosmology). We also outline some basic problems which are not yet completely solved, such as the definitions of entropy and temperature out of equilibrium, the selection of the relevant variables, and the status to be reserved to the H-theorem and its relation to the second law. In writing this review, we had four objectives in mind: to show (i) that extended irreversible thermodynamics stands at the frontiers of modern thermodynamics; (ii) that it opens the way to new and useful applications; (iii) that much progress has been achieved during the last decade, and (iv) that the subject is far from being exhausted.

https://iopscience.iop.org/article/10.1088/0034-4885/62/7/201/pdf

Extended irreversible thermodynamics revisited (1988-98)

Phonon hydrodynamics and its applications in nanoscale heat transport

A classification is given of equations vtt=f(v,vx)vxx+g(x,vx) admitting an extension by one of the principal Lie algebra of the equation under consideration. The paper is one of few applications of a new algebraic approach to the problem of group classification: the method of preliminary group classification. The result of the work is a wide class of equations summarized in Table II.

Preliminary group classification of equations vtt=f(x,vx)vxx+g(x,vx)

In this paper the complete Lie group classification of a non-linear wave equation is obtained. Optimal systems and reduced equations are achieved in the case of a hyperelastic homogeneous bar with variable cross section.

Symmetry classification and optimal systems of a non-linear wave equation

We consider a class of second‐order partial differential equations which arises in diffusion phenomena and, following a new approach, we look for a Lie invariance classification via equivalence transformations. A class of exact invariant solutions containing an arbitrary function is obtained.

A group analysis approach for a nonlinear differential system arising in diffusion phenomena

Self-adjointness of a generalized Camassa–Holm equation

This paper concerns the infinitesimal group analysis for a second order non-linear wave equation involving non-homogeneous processes. In the first part we characterize the most general expression for the generator of the Lie group. Then we calculate the invariant surfaces in some special cases, obtaining the corresponding ordinary differential equations whose integration allow us to get classes of solutions for the original equation.

A. Valenti

Papers

Preliminary group classification of equations vtt=f(x,vx)vxx+g(x,vx)

Symmetry classification and optimal systems of a non-linear wave equation

A group analysis approach for a nonlinear differential system arising in diffusion phenomena

Self-adjointness of a generalized Camassa–Holm equation

Group properties and invariant solutions for infinitesimal transformations of a non-linear wave equation☆