If you take two ideal gases at different temperatures, and allow them to share energy through heat, they'll eventually reach a thermodynamic equilibrium state that has higher entropy than the original. The time evolution of that entropy increase is easy to predict, because we know the time evolution of $T$ and $U$ for each of the gases (assuming we have the necessary constants). This is OK.
Now take a system beyond the scope of thermodynamics. A single box containing a gas that is not in thermodynamic equilibrium (doesn't follow the Boltzman distribution). One would expect that gas to quickly thermalise, and reach said equilibrium. Entropy can still be defined using statistical mechanics, and the final state will have higher entropy than the initial state.
I'm looking for quantitative experimental evidence of this effect.
Ideally, I'd like references measuring the time a gas takes to thermalise.
Obviously this time depends on many factors and is not always doable. I wasn't more specific because I don't want to be picky, I'm looking for any experiments that verify it.