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1993 | Book | 1. edition

Early Version of Extended Thermodynamics and Kinetic Theory of Gases Read chapter Read first chapter

Extended Thermodynamics

Authors: Ingo Müller, Tommaso Ruggeri

Publisher: Springer US

Book Series : Springer Tracts in Natural Philosophy

Included in: Professional Book Archive

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About this book

Physicists firmly believe that the differential equations of nature should be hyperbolic so as to exclude action at a distance; yet the equations of irreversible thermodynamics - those of Navier-Stokes and Fourier - are parabolic. This incompatibility between the expectation of physicists and the classical laws of thermodynamics has prompted the formulation of extended thermodynamics. After describing the motifs and early evolution of this new branch of irreversible thermodynamics, the authors apply the theory to mon-atomic gases, mixtures of gases, relativistic gases, and "gases" of phonons and photons. The discussion brings into perspective the various phenomena called second sound, such as heat propagation, propagation of shear stress and concentration, and the second sound in liquid helium. The formal mathematical structure of extended thermodynamics is exposed and the theory is shown to be fully compatible with the kinetic theory of gases. The study closes with the testing of extended thermodynamics through the exploitation of its predictions for measurements of light scattering and sound propagation.

Table of Contents

Frontmatter
Chapter 1. Early Version of Extended Thermodynamics and Kinetic Theory of Gases
Abstract
The laws of Fourier, Fick, and Navier-Stokes were proposed by ad hoc arguments to describe the phenomena of heat conduction, diffusion, and shear stresses in fluids. This was before there was any non-equilibrium thermodynamics. It was much later—essentially when Eckart’s papers [1], [2], [3] appeared—that these laws became incorporated into thermodynamics of irreversible processes. That theory, while not entirely systematic, is simple and has great suggestive value. We start out with a brief description of how thermodynamics of irreversible processes arrives at the laws of Fourier and Navier-Stokes.
Ingo Müller, Tommaso Ruggeri
Chapter 2. Extended Thermodynamics of Mon-Atomic Gases
Abstract
The new version of extended thermodynamics to be described in this chapter is based on Rational Thermodynamics whereas the old version is based on TIP. There are several ingredients that make rational thermodynamics more powerful than TIP, viz.
Ingo Müller, Tommaso Ruggeri
Chapter 3. Thermodynamics of Mixtures of Fluids
Abstract
The paradoxa of heat conduction and shear wave propagation described in Chapter 1 are not alone. A similar phenomenon occurs in diffusion as we shall demonstrate in this chapter. This fact calls for the formulation of an extended thermodynamic theory of mixtures.
Ingo Müller, Tommaso Ruggeri
Chapter 4. Relativistic Extended Thermodynamics of Mon-Atomic Gases
Abstract
Relativists have shown a keen interest in the development of extended thermodynamics, because the paradoxa of infinite speeds implied by the parabolic equations of ordinary thermodynamics disturbed them acutely. Indeed immediately after the formulation of the early version of extended thermodynamics Müller [5] formulated the theory relativistically, but this work was not properly published until Israel [33] presented a very similar theory. These theories had all the shortcomings of extended TIP that were discussed in Chapter 1.
Ingo Müller, Tommaso Ruggeri
Chapter 5. Thermodynamics of Light and Sound
Abstract
Light in a cavity and sound in a solid may be considered as gases of quasi-particles, the photons, and phonons respectively. It is then possible to treat them thermodynamically. The first successful attempt in that direction was Boltzmann’s derivation of what we now call the Stefan-Boltzmann law. That law relates the energy density of cavity radiation in equilibrium with the wall to the temperature of the wall.
Ingo Müller, Tommaso Ruggeri
Chapter 6. Formal Structure of Extended Thermodynamics
Abstract
Reflecting on the essentials of extended thermodynamics one realizes that the following ingredients are important:
  • basic equations of balance type;
  • constitutive equations local in space-time, hence quasilinear first-order field equations;
  • entropy inequality with concave entropy density, hence symmetric hyperbolic field equations.
Ingo Müller, Tommaso Ruggeri
Chapter 7. Waves in Extended Thermodynamics
Abstract
Waves represent an important probe into the constitutive properties of bodies, because their speeds and attenuations may be measured as well as, in the case of shock waves, the jumps in density, temperature, etc. Such observations permit the calculation of constitutive coefficients or alternatively, when the coefficients are known, they permit the evaluation of the range of validity of the theory.
Ingo Müller, Tommaso Ruggeri
Chapter 8. Second Sound in Solids at Low Temperature
Abstract
The phenomenon of second sound in liquid He II was first predicted in the theoretical works of Tisza [78] and Landau [79]. Later Peshkov [80] detected this wave and confirmed that its temperature variations are uncoupled from the pressure variations: therefore the second sound is often referred to as a temperature wave. This phenomenon is now well understood. We described it in Chapter 3 following the work of Dreyer [81], who showed that the bifluid theory of Tisza and Landau is a special case of the thermodynamic theory of binary mixtures in general.
Ingo Müller, Tommaso Ruggeri
Chapter 9. Molecular Extended Thermodynamics
Abstract
On previous occasions we have been surprised that extended thermodynamics furnished the same results as the kinetic theory of gases in the Grad approximation. The present chapter shows that this analogy goes far beyond 13 moments. Indeed, if we make sure that the phenomenological and the molecular theories choose the same variables, irrespective of their number, we can prove that extended thermodynamics and molecular extended thermodynamics lead to identical results. This was first shown by Dreyer [95]. To be sure, the methods are different in the two fields and, in particular, the role of entropy is different: In extended thermodynamics the entropy inequality is exploited, while in the molecular theory we maximize the kinetic expression for the entropy.
Ingo Müller, Tommaso Ruggeri
Chapter 10. Testing Extended Thermodynamics by Light
Abstract
Light scattering is a powerful tool for the experimental determination of transport coefficients like heat conductivity and viscosity. The great advantage of this method is that it works in equilibrium; in fact, it makes use of the density fluctuations in equilibrium. By Onsager’s hypothesis the auto-correlation function of such a fluctuation is related to the solution of the macroscopic field equations.
Ingo Müller, Tommaso Ruggeri
Chapter 11. Testing Extended Thermodynamics by Sound
Abstract
Experimentalists measure phase speeds and attenuations in acoustic resonators. On the other hand, any thermodynamic theory will provide a mathematical expression for the phase speeds and attenuations. If the theory contains unknown parameters, a comparison of the empirical data with the theoretical prediction delivers values for these parameters; but, if a theory is specific—as is the case with molecular extended thermodynamics—the experiments provide an estimate of the range of validity of the theory.
Ingo Müller, Tommaso Ruggeri
Backmatter
Metadata
Title
Extended Thermodynamics
Authors
Ingo Müller
Tommaso Ruggeri
Copyright Year
1993
Publisher
Springer US
Electronic ISBN
978-1-4684-0447-0
Print ISBN
978-1-4684-0449-4
DOI
https://doi.org/10.1007/978-1-4684-0447-0