Entropy is incredibly useful as a mathematical tool. But what does it actually mean? I understand that the Boltzmann entropy is defined by:
With $\Omega$ being the multiplicity of the system. As pointed out by many other QA pairs on this site, this mathematical definition is highly favorable and helps simplify calculations.
But beyond being just a helpful mathematical tool, does entropy have a physical interpretation? I have heard a few explanations but they all seem to fall short. For example,
1) Entropy is a measure of disorder. 2) Entropy is a measure of heat flow. 3) Entropy is a measure of energy.
My problems with each of these I hope are quite reasonable.
1) "Disorder" is a completely ambiguous term. If I don't clean up my house for a year it will certainly be disordered but it has not been evolving exploring every possible macrostate and thus the entropy hasn't been increasing, rather it has stayed at zero.
2) Put a gas in a sealed box with an attached vacuum chamber. Remove the partition and allow the gas to fill both chambers. The entropy of the system has increased and yet no heat flowed into/out of the system.
3) The units of $k$ are $J/K$. This doesn't make sense based on unit analysis alone.
What I'm looking for is one of two things. A concrete example of the physical interpretation of entropy or a solid statement that can explain why it has no physical interpretation and is a pure statistical phenomenon.