Relativistic hydrodynamics: gradient expansion

AI Thread Summary
Relativistic anisotropic hydrodynamics involves a "gradient expansion" of the energy-momentum tensor, which is crucial for deriving the Boltzmann transport equation from off-equilibrium quantum-many-body theory. This expansion requires starting from local thermal equilibrium and performing a second-order gradient expansion to avoid acausality in the relativistic case, leading to the Israel-Stewart hydro equation. Key literature on this topic includes works by Denicol et al., which provide systematic approaches to moment expansions. The discussion also highlights the importance of foundational texts, such as the recently published book by Paul Romatschke, for those new to hydrodynamics. Overall, the conversation emphasizes the need for a solid understanding of gradient expansions in the context of relativistic hydrodynamics.
klabautermann
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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!
 
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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.,

https://arxiv.org/abs/1004.5013https://arxiv.org/abs/1202.4551https://arxiv.org/abs/1206.1554
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!
 
Thanks for the tip, I'm certainly going to look into it!
 
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