Relativistic Extended Thermodynamics Research Articles

On two possible ways to recover ordinary thermodynamics from extended thermodynamics of polyatomic gases

We consider two possible ways, i.e. the Maxwellian iteration (MI) and the Chapman–Enskog method (CEM), to recover relativistic ordinary thermodynamics from relativistic extended thermodynamics of Polyatomic gases with N moments. Both of these methods give the Eckart equations which are the relativistic version of the Navier–Stokes and Fourier laws as a first iteration. However, these methods do not lead to the same expressions of the heat conductivity χ, the shear viscosity µ, and the bulk viscosity ν which appear as coefficients in the Eckart equations. In particular, we prove that the expressions of χ, µ, and ν obtained via the CEM do not depend on N, while those obtained through the MI depend on N. Moreover, we also prove that these two methods lead to the same results in the nonrelativistic limit.

Hyperbolicity of a model for polyatomic gases in relativistic extended thermodynamics

The hyperbolicity of a model for relativistic extended thermodynamics of polyatomic gases is here proved for every time direction $$\xi _\alpha $$ ; moreover, an expression for the production terms of the field equation is proposed which encloses as particular cases the variant of the Anderson–Witting model already known in the literature, but also a new variant of the Marle model. With these expressions, it is shown that the field equations satisfy automatically the requirement of a non-negative entropy production.

A 16 moments model in relativistic extended thermodynamics of rarefied polyatomic gas

A model with 16 moments is here presented in the framework of RET of polyatomic gases. It furnishes as principal subsystem the relativistic counterpart of a work by Arima T., Ruggeri T., Sugiyama M.; this is present in literature and treats the non relativistic case which incorporates relaxation processes of molecular rotation and vibration. Another principal subsystem is the natural extension of the 14 moments model by Pennisi S. and Ruggeri T.; this is also present in literature in the relativistic framework but where the trace of the third balance equation is neglected. Its extension is found here for the case when this trace isn’t neglected.

Production terms in relativistic extended thermodynamics of gas with internal structure via a new BGK model

The goal of this paper is to obtain a precise expression for the production tensor in a dissipative hyperbolic relativistic theory of gas with internal structure. For this aim, we use a variant of relativistic BGK model for the Boltzmann–Chernikov kinetic equation. Moreover, we deduce some inequalities for the coefficients requiring the entropy principle and the convexity of entropy density. As a limiting case, the expression of the production tensor in the case of monatomic gas is also evaluated.

Monatomic limit of relativistic extended thermodynamics of polyatomic gas

In a recent paper Pennisi and Ruggeri (Ann Phys 377:414–445, 2017. https://doi.org/10.1016/j.aop.2016.12.012 ) proposed a casual hyperbolic model for a dissipative polyatomic relativistic gas. The closure was obtained using the maximum entropy principle for the generalized moments of a distribution function that, as in the classical case, depends on an additional continuous variable representing the energy of the internal modes of a molecule; this permits the theory to take into account the energy exchange between translational modes and internal modes of a molecule. The closure depends on a parameter $$a>-1$$ that is related to the degrees of freedom of gas. In this paper it is proven that in the singular limit for $$a \rightarrow -1$$ the field equations converge to the system obtained by relativistic extended thermodynamics theory of monatomic gas by Liu et al. (Ann Phys 169:191, 1986).

Integrability properties for relativistic extended thermodynamics of polyatomic gas

In a recent article (Ann Phys 377:414–445, 2017. https://doi.org/10.1016/j.aop.2016.12.012 ), Pennisi and Ruggeri proposed a relativistic extended thermodynamics theory of polyatomic gas. It was achieved by adopting the closure procedure for the generalized moments of a distribution function that, as in the classical case, depends on an additional continuous variable representing the energy of the internal modes of a molecule; this permits the theory to take into account the energy exchange between the translational and the internal modes of a molecule in binary collisions. In this paper the attention will be focused on some integrals appearing in the field equations of the relativistic theory of polyatomic gas and their integrability will be proven. In the last part of the paper we consider the case of a monatomic gas and we evaluate the ultra-relativistic limit of the differential system of balance laws.

Relativistic extended thermodynamics of rarefied polyatomic gas

The goal of the present paper is to construct a relativistic extended thermodynamics (ET) theory of rarefied polyatomic gases. This is achieved by adopting the closure procedure for the 14 generalized moments of a distribution function that, as in the classical case, depends on an additional continuous variable representing the energy of the internal modes of a molecule. This permits the theory to take into account the energy exchange between translational modes and internal modes of a molecule in binary collisions. The theory includes the ET theory of monatomic gases as a singular limit and converges to the corresponding ET theory of polyatomic gases in the classical limit.In contrast to a monatomic gas in which the order of magnitude of the dynamical pressure is O(1/c2), the dynamical pressure in a polyatomic gas is not small due to the internal motion of molecules as is the case in the classical limit. Therefore the present theory might result particularly useful in cosmology to describe some aspects of the post-recombination era.

Relativistic extended thermodynamics from the Lagrangian view-point

The salient points of relativistic extended thermodynamics (R.E.T.) are here revised according to the Lagrangian view-point: attention is focused to each material particle and to its physical properties, during all the motion of the same particle. The results for the non relativistic case are already present in literature. Here a similar procedure is followed for R.E.T. with an arbitrary number of moments. It is also shown how the Einstenian Relativity Principle and some symmetry condition, which are present in the Eulerian view-point, can be “translated” in the Lagrangian view-point, where they are no more so self-evident.

The model with many moments for relativistic electron beams: A simplified solution

In the 1980s, Amendt and Weitzner proposed an interesting model capable to describe relativistic electron beams. It concerned 14 independent variables and the closure was obtained by using the entropy and the Einstein relativity principles. As we know from literature, an extension to many moments allows to achieve an improvement in the results. Three years ago, we exhibited a macroscopic model with an arbitrary but fixed number of moments for relativistic extended thermodynamics. Such model was more general than those previously appeared in literature, so it was applicable even to materials different from an electron beam. Subsequently, we found the closure of such model consistent with the entropy and the Einstein relativity principles, up to whatever order with respect to equilibrium. The solution was determined in terms of a family of arbitrary single variable functions arising from integration. Those results have a very complex shape and are very difficult to handle so a simplification is necessary. In this paper we will reach this goal. Furthermore, we will prove that by fixing a certain order np with respect to equilibrium and a scalar valued single variable function, appearing at that order, then all the terms appearing at orders n ⩽ np are determined without introducing any other function. This result has already been found for the nonrelativistic case but its extension to the relativistic framework is not straightforward and it requires a supplementary mathematical tool: the above mentioned simplification in the shape of the solutions.

On a generalized kinetic approach to Relativistic Extended Thermodynamics

Extended Thermodynamics is applicable to many physical problems. Hydrodynamics models are also capable to describe the charge carrier transport in semi conductors, particularly with regard to hot electrons and submicron devices. In a recent paper [1] we considered the macroscopic model with an arbitrary but fixed number of moments for Relativistic E.T. and we found the closure consistent with the entropy principle and the Einstein relativity one, up to whatever order with respect to equilibrium. Everything was determined in terms of an arbitrary family of a single variable functions arising from integration. Subsequently, in paper [2], we proved that fixing a certain order np with respect to equilibrium and a scalar valued single variable function ψ, appearing at that order, then all the other terms appearing at orders n ≤ np are determined in terms of ψ, without introducing any other function. In this paper we will prove that the same result can be found also with a generalized kinetic approach. It has the advantage of being expressed in a more compact way. 1. M. Carrisi and S. Pennisi, An exact fluid model for relativistic electron beams: The many moments case, Journal of Math. Phys., vol. 48, no. 12, p. 123101, 2007. 2. M. Carrisi and S. Pennisi, A simplified exact macroscopic approach to relativistic extended thermodynamics, with an arbitrary number of moments, to be published.

Entropy principle and relativistic extended thermodynamics: Global existence of smooth solutions and stability of equilibrium state

The entropy principle plays an important role on hyperbolic systems of balance laws: symmetrization, principal subsystems and nesting theories. equilibrium manifold. After a brief survey on these questions we present somerecent results concerning the local and global well-posedness of the Cauchy problem for smooth solutions with particular attention to the genuine coupling Kawashima condition. These results are applied to the case of Relativistic Extended Thermodynamics and we prove that there exist global smooth solutions for small initial data and the solutions converge to a constant equilibrium state.

An Increasing Number of Moments in Relativistic Extended Thermodynamics

Thermodynamic models with an increasing number of moments are here compared, in the frame-work of relativistic extended thermodynamics. They constitute a family of models depending on a parameter n. In these models, some constitutive functions occur; a part of them are determined by imposing that they satisfy exactly the entropy and the relativity principles. It is pleasant to see how many elegant symmetries, which are present in the pioneering work by Liu, Muller and Ruggeri about the 14 fields model, nicely extend also to the general case.

Speeds of Propagation in Classical and Relativistic Extended Thermodynamics.

The Navier-Stokes-Fourier theory of viscous, heat-conducting fluids provides parabolic equations and thus predicts infinite pulse speeds. Naturally this feature has disqualified the theory for relativistic thermodynamics which must insist on finite speeds and, moreover, on speeds smaller than c. The attempts at a remedy have proved heuristically important for a new systematic type of thermodynamics: Extended thermodynamics. That new theory has symmetric hyperbolic field equations and thus it provides finite pulse speeds.Extended thermodynamics is a whole hierarchy of theories with an increasing number of fields when gradients and rates of thermodynamic processes become steeper and faster. The first stage in this hierarchy is the 14-field theory which may already be a useful tool for the relativist in many applications. The 14 fields — and further fields — are conveniently chosen from the moments of the kinetic theory of gases.The hierarchy is complete only when the number of fields tends to infinity. In that case the pulse speed of non-relativistic extended thermodynamics tends to infinity while the pulse speed of relativistic extended thermodynamics tends to c, the speed of light.In extended thermodynamics symmetric hyperbolicity — and finite speeds — are implied by the concavity of the entropy density. This is still true in relativistic thermodynamics for a privileged entropy density which is the entropy density of the rest frame for non-degenerate gases.

Matter creation and bulk viscosity in early cosmology

The two distinct irreversible phenomena of matter creation and bulk viscosity are considered as independent, giving cross effects in relativistic extended irreversible thermodynamics, as applied to the primordial Friedman-Lemaître-Robertson-Walker cosmology. Some consequences are developed in the isentropic exponential inflation case.

Relationship between two theories of dissipative relativistic hydrodynamics applied to cosmology.

We compare Israel's relativistic causal theory of transient effects with the relativistic extended irreversible thermodynamics of Pav\'on, Jou, and Casas-Vasqu\`ez, applying both to the case of cosmic fluid with bulk viscosity only. They are completely equivalent up to the first order in the parameter characterizing the departure from equilibrium. We show, if certain relations between the main variables of the two theories hold, that they are also equivalent up to the second order in this parameter. We also extend up to any order Israel's theory and give the relations permitting again the equivalence between the two theories in this general case. Some cosmological consequences are discussed.

Extended thermodynamics of Friedmann-Robertson-Walker models in the Landau-Lifshitz frame

The authors study dissipative Friedmann-Robertson-Walker (FRW) models with the help of the relativistic extended thermodynamics as formulated in the Landau-Lifshitz (energy) frame. The physically most relevant solutions are shown to tend asymptotically to models in which the initial material anisotropy due to the particle drift (diffusion) is washed out. Some recent results obtained by Turok (1988) and Barrow (1988) in a different context can be recovered.

Wave propagation in relativistic extended thermodynamics of a charged fluid

An analysis of discontinuity wave propagation across a constant state for a relativistic charged fluid with finite electric conductivity is performed. Beside thecontact wave usual in fluid dynamics, acharge wave arises which propagates with the drift speed of the ions. The remaining eight waves are all coupled together and reduce to magnetoacustic and Alfven waves when the electric conductivity tends to infinity, while they become purely electromagnetic waves when the electric permutivity and the magnetic permeability of the medium are the same as in vacuo.

Fluid models for relativistic electron beams: An independent derivation

A generalized fluid model is proposed for a relativistic charged particle beam. The model is constructed by starting with the covariant Vlasov equation, taking the moments, and closing via a suitable ‘‘constitutive law’’ for the third-order moment. The ‘‘constitutive law’’ is uniquely specified by requiring consistency with a supplementary conservation law representing entropy balance. The derivation presented here is independent and also different from a previous one utilizing results obtained within the framework of relativistic extended thermodynamics.

Wave propagation in relativistic Extended Thermodynamics

We classify the thermodynamical waves that propagate in a non-equilibrium gas governed by the field equations of relativistic extended thermodynamics. The calculations are carried out for a generic state function appropriate to mon-atomic ideal gases. In the particular cases of the non-degenerate gas specific wave speeds are calculated. In the limit of the non-relativistic and the ultra-relativistic gas the results agree with those given before in literature.

Convexity and symmetrization in relativistic theories

There is a strong motivation for the desire to have symmetric hyperbolic field equations in thermodynamics, because they guarantee well-posedness of Cauchy problems. A generic quasi-linear first order system of balance laws — in the non-relativistic case — can be shown to be symmetric hyperbolic, if the entropy density is concave with respect to the variables. In relativistic thermodynamics this is not so. This paper shows that there exists a scalar quantity in relativistic thermodynamics whose concavity guarantees a symmetric hyperbolic system. But that quantity — we call it — $$\bar h$$ — is not the entropy, although it is closely related to it. It is formed by contracting the entropy flux vector — ha with a privileged time-like congruence $$\bar \xi _\alpha $$ . It is also shown that the convexity of h plus the requirement that all speeds be smaller than the speed of light c provide symmetric hyperbolic field equations for all choices of the direction of time. At this level of generality the physical meaning of —h is unknown. However, in many circumstances it is equal to the entropy. This is so, of course, in the non-relativistic limit but also in the non-dissipative relativistic fluid and even in relativistic extended thermodynamics for a non-degenerate gas.