Abstract
In this chapter we make an important departure from the approach taken in Chaps. 3, 5, 6 and 7 of Volume 1 in which nonrelativistic kinetic equations have been discussed for gases and liquids. Now we consider relativistic kinetic equations for dilute uncorrelated particle systems.
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Notes
- 1.
The sign of the projector defined here is opposite to the projector used in Ref. [4].
- 2.
- 3.
This aspect seems to indicate that the vacuum may be described by hydrodynamic equations in the limit of \(u/c\rightarrow 0\) in relativity.
- 4.
- 5.
It should be noted and kept in mind that the projector \(\Delta _{\sigma }^{\mu }\), etc. are to be applied after the statistical mechanical averaging is performed on the molecular expression of the moment. Therefore, if one wishes to understand, for example, \(h_{a}^{(1)\mu \nu }\) at the molecular level the factor
$$ \left( \Delta _{\sigma }^{\mu }\Delta _{\tau }^{\nu }-\frac{1}{3}\Delta ^{\mu \nu }\Delta _{\sigma \tau }\right) p_{a}^{\sigma }p_{a}^{\tau } $$should be taken as
$$ p_{a}^{\mu }p_{a}^{\nu }-\frac{1}{3}g^{\mu \nu }\left( p_{a}^{\sigma }p_{a\sigma }\right) , $$where \(\left( p_{a}^{\sigma }p_{a\sigma }\right) \) is the trace of \(p_{a}^{\sigma }p_{a}^{\tau }\), which is equal to \(m_{a}c^{2}\). Here traceless symmetry generating operator \(\left( \Delta _{\sigma }^{\mu }\Delta _{\tau }^{\nu }-\frac{1}{3}\Delta ^{\mu \nu }\Delta _{\sigma \tau }\right) \) is meant to be applied after statistical mechanical averaging is taken.
- 6.
According to the Boltzmann kinetic theory, i.e., nonrelativistic kinetic theory, \(h_{a}^{(3)\mu }=\frac{1}{2}m_{a}\left( \mathbf {C}_{a}\cdot \mathbf {C}_{a}\right) C_{a\,\mu }-\widehat{h}J_{a\,\mu }\), where \(\mathbf {C}_{a}\) is the peculiar velocity \(\mathbf {C}_{a}=\mathbf {v} _{a}-\mathbf {u}\).
- 7.
The set \(\left\{ h_{a}^{(q)}\right\} \) must be closed such that the thermodynamic branch \(f_{a}^{\text {c}}\left( x,p_{a}\right) \) of the distribution function \(f_{a}\left( x,p_{a}\right) \) is normalizable, that is, the set must be closed at the even order of \(h_{a}^{(q)}\) so that the integral involved is convergent. See (1.210) below for \(f_{a} ^{\text {c}}\left( x,p_{a}\right) \).
- 8.
The proof follows the same procedure as for the nonrelativistic Boltzmann collision integral. For this reason it is not shown here to avoid repetition.
- 9.
This point can be better comprehended if we recall that the distribution function \(f_{a}(x,p_{a})\) is a singlet distribution function of the many-particle distribution function representing the ensemble of representative systems.
- 10.
As a matter of fact, at this point \(T\left( x\right) \) here is not known to be the temperature field. We are anticipating it to be the temperature of the equilibrium system when thermodynamic correspondence is made according to the thermodynamic theory of processes as will be shown later. Thus we may set \(T_{\text {e}}=T\left( x\right) \).
- 11.
In this form the normalization factor for \(f_{a}^{\text {e}}\) is given by
$$ n_{a}^{\text {e}}\exp \left( -\beta _{\text {e}}\mu _{a}^{\text {e}}\right) /\left\langle \exp (-\beta _{\text {e}}U_{\nu }p_{a}^{\nu }\right\rangle . $$ - 12.
In the approach taken in the present work the thermodynamic correspondence is taken alternatively to Tolman’s equipartition law.
- 13.
This means that the distribution function is given in the units of action cubed, e.g., \(h^{3} \), where h is the Planck constant. This requires that the other averages defined earlier are in the units of \(h^{3}\), the dependence of which can be appropriately later. Provided we remember this adjustment in dimension for them there is no harm done even if the formulation is carried out without the factor \(h^{3}\) in the normalization factor.
- 14.
Since the phenomenological theory of irreversible relativistic processes is not established, the theory formulated here is the reverse of that of the nonrelativistic theory taken in the nonrelativistic chapter. Here by extending the nonrelativistic theory into the relativistic domain it may be said that a relativistic theory of thermodynamics of irreversible processes is formulated.
- 15.
It may be called the complement to the projection onto the manifold \(\mathfrak {P\cup T}\) of \(f_{a}\left( x,p_{a}\right) \).
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Eu, B.C. (2016). Relativistic Kinetic Theory for Matter. In: Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-41153-8_1
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