Abstract
The relativistic kinetic theory of irreversible processes in a system of matter presented in Chap. 1 of this volume is generalized to include radiation in this chapter.
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- 1.
This chapter is a substantially revised version of the unpublished paper by the author with K. Mao, which was also part of the PhD thesis of K. Mao, McGill University, Montreal, 1993, under supervision of B.C. Eu. The revision is concerned with the theory of irreversible thermodynamics making use of the calortropy and the evolution equations for nonconserved variables. There are other aspects which are significantly revised.
- 2.
Later, this statement is shown to be true from the thermodynamic consideration.
- 3.
This is a covariant kinetic equation for particles free from an external force field \(F^{\mu }\). This restriction, however, can be relaxed for the case of the Lorentz force, provided that the external force leaves the rest mass of the particle unaltered at the end of the particle collision
$$ p^{\mu }F_{\mu }=0. $$Furthermore, it is assumed that the particles having momenta in the range \(\Delta ^{3}p\) at the beginning of the proper time interval \(\Delta \tau \) occupy an equal momentum range at the end of the interval. This would true if the force has the property
$$ \frac{\partial }{\partial p^{\mu }}F^{\mu }(x,p)=0. $$These properties are met by the Lorentz force. For such an external force the covariant kinetic equation may be modified to the form
$$ p_{a}^{\nu }\partial _{\nu }f_{a}(x,p_{a})+mF_{a}^{\mu }f_{a}(x,p_{a} )=\mathfrak {R}_{a}\left[ f_{a}\right] , $$where \(F_{a}^{\mu }\) is external force on particle a.
- 4.
Here we would like to emphasize that the hydrodynamic velocity is defined according to Eckart, but for the mixture of matter and photons:
$$ U^{\mu }=\frac{cN^{\mu }}{\sqrt{N^{\nu }N_{\nu }}}, $$where \(N^{\mu }\) is the sum of \(N_{i}^{\mu }\) with the index i running for entire species including photons according to our model.
- 5.
This form is without a constant factor that has to do with the normalization of \(f_{a}^{\text {e}}\). See Chap. 1 where the case for classical particles is described in connection with equilibrium distribution functions.
- 6.
It should be noted that \(\mathfrak {p}^{\text {e}}v_{a}\) on the left of (2.36) defines macroscopic parameter \(\mathfrak {p} ^{\text {e}}\) by the statistical mechanical formula on the right, which will turn out to be hydrostatic (i.e., equilibrium) pressure, when \(\mathcal {S} ^{\text {e}}\) calculated therewith is identified with the equilibrium entropy on correspondence with the phenomenological Clausius entropy deduced from the second law of thermodynamics.
- 7.
- 8.
The solution is approximate in the sense that it is given by a projection of the phase space distribution function onto the thermodynamic manifold of macroscopic variables whose dimension is much smaller than the full phase space of the system.
- 9.
This is also for the case of matter consisting of a gas.
- 10.
For higher rank tensors isotropic tensors of higher rank, more complicated basis sets would be required. For the basis sets for higher order isotropic Cartesian tensors, see pages 97–98, Chap. 5, Ref. [18].
- 11.
It should be noted here that we have not eliminated the dependent diffusion flux in view of the fact that the number and range of photon spectrum is indeterminate in the present model.
- 12.
For the detail of getting \(q_{L}(\kappa _{L})\) from the nonlinear factor \(q_{n}(\kappa )\) and \(\kappa _{L}\), see Chap. 1 of this Volume.
- 13.
For example, de Groot et al. [29] define the Boltzmann entropy four-flow by the formula
$$ S^{\mu }\left( x\right) =-k_{B}c\sum _{a=1}^{\left\{ m,r\right\} }\int \frac{d^{3}p_{a}}{p_{a}^{0}}p_{a}^{\mu }f_{a}\left( \ln f_{a}-1\right) . $$We define \(S^{\mu }\left( x\right) \) by
$$ S^{\mu }\left( x\right) =-k_{B}c\sum _{a=1}^{\left\{ m,r\right\} }\int \frac{d^{3}p_{a}}{p_{a}^{0}}p_{a}^{\mu }f_{a}\ln f_{a} $$without the \(\left( -1\right) \) factor, which we find superfluous.
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Eu, B.C. (2016). Relativistic Kinetic Theory of Matter and Radiation. In: Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-41153-8_2
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