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Wiki states

While the classic Maxwell-Juttner distribution generalizes for the case of special relativity, it fails to consider the anisotropic description.

What are the challenges when one tries to build an anisotropic relativistic distribution for a gas? Why can't one just start with isotropic description and apply a boost?

More Anonymous
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  • I am very skeptical of that whole paragraph. (1) There is no reference to an external source to justify the claims made. (2) The previous sentence in the same paragraph says "the basic MB [Maxwell-Boltzmann] distribution... assumes anisotropic temperature" which is incorrect. (3) Looking at the revision history, that paragraph, and a lot of content on the page, was added by one user who no longer has a wikipedia account, and some of that user's edits were reverted for "possible vandalism." – Andrew Aug 03 '22 at 00:14
  • @Andrew it seems paper are written on this topic (I don't have access since it's behind the paywall). https://ui.adsabs.harvard.edu/abs/2016AnGeo..34.1145L/abstract – More Anonymous Aug 03 '22 at 03:28
  • That paper doesn't contradict the sentence on wikipedia though. I interpret the wikipedia article as saying that the classic Maxwell-Juttner distribution is isotropic. Therefore it makes sense that people write papers attempting to construct an anisotropic version. However, my main point is that I would not recommend using the wikipedia article as a starting point for learning about the MJ distribution, because there are some things in that article that seem suspicious to me. If you can find a more reliable source (textbook, peer-reviewed review article) I would start there. – Andrew Aug 03 '22 at 11:03
  • Very related: https://physics.stackexchange.com/q/216819/59023 – honeste_vivere Aug 08 '22 at 20:36

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I think the main issue is that this is a needlessly confusing wikipedia entry, apparently written by someone who is trying to promote his own paper on anisotropic distributions.

There is a unique relativistic generalization of the Boltzmann distribution that describes the distribution function of a non-interacting gas in perfect local equilibrium. This is the Juttner distribution. Of course, real systems interact, and are not in perfect equilibrium. In particular, there may be a background flow that breaks the local isotropy of the distribution.

However, there is no general model for these effects. The actual distribution has to be determined from solution of the Boltzmann (or Boltzmann-Vlasov) equation that take into account the actual interaction and expansion history.

There is one limiting case in which one give an (almost) model independent result. In a weakly sheared fluid the distribution function is $$ f=f_0+\delta f = f_0\left(1 + \frac{8\Gamma_s}{T^2} p_\mu p_\nu \nabla^{\langle\mu}u^{\nu\rangle} \right) $$
where $T$ is the temperature, $\Gamma_s=4\eta/(3sT)$ with $\eta$ the viscosity and $s$ the entropy density is the sound attenuation rate, $u^\mu$ is the four-velocity of the fluid, and $A^{\langle\mu\nu\rangle}$ is the symmetric-traceless part of the tensor $A^{\mu\nu}$.

Thomas
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