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Does classical thermodynamics hold up at relativistic speeds?

Hi there, undergrad materials engineer here. I was wondering if classical thermodynamics would hold up for matter traveling at relativistic speeds. The idea behind this question started with a friend and I discussing accretion discs of black holes, and wondering how to model diffusion/mixing processes as one nears the event horizon.  Questions like how would the chemical potential for adding a proton to the accretion disk change if the matter of the disk were travelling at .9c and the proton was travelling at v << c  verses  v =.9c?
gpt
Classical thermodynamics holds for relativistic motion basically unchanged, at least if we are concerned with a single local "patch" of fluid. Because of the principle of relativity, we can describe the physics of a blob of material traveling at 0.9 c with respect to the "lab frame" in terms of its properties in a frame moving with the blob, in which it is at rest. In this frame, the velocity is zero so the 1st and 2nd law of thermodynamics are the very same equations you are used to in the non-relativistic context. The only difference for accretion disks around compact objects is that there might be thermodynamical processes (pair creation, nuclear reactions) that are more naturally accounted for by adding the rest-mass contribution to the energy in the chemical potentials (the famous m c\^2).

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If you want to describe non-local phenomena like advection and mixing then things become a little bit more involved, because we need to use Lorentz transformation to relate local quantities with quantities measured in a global frame. In other words, the standard equations for, e.g,. gasdynamics, must be replaced by their relativistic counterparts.

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Diffusion is even more complex, because the way we describe diffusion in the classical context cannot be extended in a simple way to relativity.