A piece of scientific text

При установлении термодинамического равновесия происходит взаимная диффузия

Can you help please?
установление термодинамического равновесия - thermodynamic equilibration,
but what preposition does it require?

While "thermodynamic equilibrium" is technically correct, the term "thermal equilibrium" is more common in English scientific literature. This particular phrase could be translated as "when thermal equilibrium is reached", or, more stylishly, "when thermal equilibrium is attained" or "... is established".

Can I say "DURING thermal equilibrium is being established"? Because it's important to show both processes occur in the same time

Not that I understand a word of what you're talking about BUT you'd have to say -during the establishment of thermal equilibrium - what you're put - during thermal equilibrium is being established - doesn't make sense. Write the whole sentence and it will be easier to make sure the comprehension and grammar are correct.

bad manners, BJ, thank you very much for your help

First of all, the original sentence does not specifically mean that both processes happen simultaneously. It may mean it, but it is only an interpretation.

Secondly, even if it is correct, the translated sentence does not have to be more precise than the original one.

Thirdly, if you want it to be precise, then you should re-arrange the whole thing. An example of that would be "mutual diffusion continues until thermal equilibrium is attained". This is taken out of any context so it may not be the best alternative for you.