Abstract
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.
Highlights
Relativistic thermodynamics is needed, because in relativity the mass of a body depends on how hot it is and the temperature is not necessarily homogeneous in equilibrium
By the Cauchy–Kowalewski theorem these derivatives are local representatives of an analytical thermodynamic process and the entropy principle requires that the field equations and the entropy equation must hold for all u,A
In that case there was no upper bound so that the pulse speeds tended to infinity for extended thermodynamics of very many moments
Summary
Relativistic thermodynamics is needed, because in relativity the mass of a body depends on how hot it is and the temperature is not necessarily homogeneous in equilibrium. Weiss [49], working with the non-relativistic kinetic theory of gases, demonstrated that the pulse speed increases with an increasing number of moments. The quest for macroscopic field theories with finite pulse speeds has proved heuristically useful for the discovery of the formal structure of thermodynamics, relativistic and otherwise. New insight is provided into the form of the transport coefficients: bulk- and shear-viscosity, and thermal conductivity, which are all explicitly related here to the relaxation times of the gas This whole review is concerned with a macroscopic theory: Extended thermodynamics. Thermodynamic processes are defined and characteristic speeds and the pulse speed are introduced
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