Experiments that measure the time a gas takes to reach equilibrium 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.
 A: "The time a gas takes to reach equilibrium" isn't really well-defined -- and even trying to interpret it in terms of an average sense can be tricky: in particular, there may very well be states which never reach equilibrium (consider very carefully placed and aimed gas moleculres in a box).  As far as physics is concerned though, the Boltzmann equation is what's believed to govern the approach to equilibrium, but one must put in some assumed "collision term" to actually put it to use.
In terms of where your question may have been studied in current research, the issue of thermalization is obviously going to be a bigger issue in fields of physics when one is dealing with numbers of particles near the limits of validity of thermodynamics, so that's a sign that you might look for papers in the field of "cold atoms".  See for example this 2001 paper by the group of Alain Aspect which carefully measures a "thermalization time" $\tau_{th}$ which they define via a Boltzmann equation-inspired model inspired by a theory paper of Luiten et al.  These papers might be a good place to start to see how working physicists think about thermalization.
You frame your question in terms of "ideal gases" but as you'll notice from these papers, when you get down to numbers of particles and temperatures which are manageable directly in these experiments, quantum effects play a nontrivial role in the population dynamics.  (It's also worth giving some thought to the following: how can an ideal gas equilibrate if the particles are defined not to interact with each other?  Your answer will probably help you figure out better what exactly you're looking for.)
If you're more interested in what happens to a large number of "classical" particles and how they thermalize purely through collisions, this is fairly easy to simulate on the computer with molecular dynamics code, and then you can measure whatever you like with the computed particle trajectories.  Of course this isn't strictly a physical "experiment" but it is in my opinion a real test of the kinetic theory predictions all the same.
A: I unfortunately don't have too much time to continue to research this question.  I can give direction to you for your own research.  I was somewhat intrigued by the question because the thrust of it is to imply that macroscopic physical laws are somehow invalid. To your point, classical laws do somewhat breakdown in the quantum regime do to phenomenon such as Bose-Einstein condensates:
//cua.mit.edu/ketterle_group/popular_papers/physics%20today%20v2.pdf
Which is obviously an area of active research.
However, in general your question can be alternatively be understood as a test of diffusivity governed by the Arrhenius equation.  The Science and Engineering of Materials Second Edition by Donald Askelund, chapter 5 on the Atom Movement of Materials, discuss Arrehenius equation in the context of movement of defects and imperfections in materials.  In the context provided though, one would assume that the vacuum surrounding an initial lump of gas as being a material (no I do not prescribe to aether, but this is a convenient assumption). So the question then is one of what the activation energy is to cause the atoms to move into the vacuum vs the potential energy available for such a reaction.
A quick look for test of the Arrhenius equation will reveal some intersting results.
As you move further into this question you find discussion on reaction and diffusivity:
//www.scholarpedia.org/article/Reaction-diffusion_systems
There are multiple examples of diffusion theory underlying the processes used in several areas of technology.  So I'm not sure about the context and intent of your question.
