I Relativistic hydrodynamics: gradient expansion

Hi everyone,
I'm interested in relativistic anisotropic hydrodynamics and often a "gradient Expansion" is mentioned in articles, but not how this works exactly. I gathered that this is some kind of expansion of the energy-momentum tensor. Can someone explain to me how this expansion is set up, or better yet point me to some literature or articles?

Thanks a lot!
 

vanhees71

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The gradient expansion usually refers to the derivation of the Boltzmann transport equation from off-equilibrium quantum-many-body theory.

Now having the Boltzmann equation to get hydrodynamics you start from an expansion of the collision term around local thermal equilibrium and do another gradient expansion. For the relativistic case you must go to at least the 2nd order in gradients since Navier-Stokes is acausal in the relativistic case. In the most simple version this leads to the Israel-Stewart hydro equation. For a systematic way in terms of moment expansions, see the works by Denicol et al, e.g.,


Maybe I can give more specific references, if you tell me, which paper(s) you are referring to.
 
Good evening,

first of all, thanks for your reply and the references. Actually I need to understand this https://arxiv.org/abs/1712.03282 . Since I had virtually no exposure to hydrodynamics so far, I am trying to get a general picture about the subject, so I am working through this https://arxiv.org/abs/0902.3663 right now. For my purposes, its probably not all that necessary to know, but, I have to say, I am getting more and more intrigued by the subject. So I just wanted know.

At first sight the papers you posted seem to be exactly what I was looking for, thanks again!
 

vanhees71

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Great! Note that Paul Romtaschke has written a very nice textbook on the subject, which just came out at Cam. Uni. Press:

 
Thanks for the tip, I'm certainly going to look into it!
 

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