The same microstate will have different probabilities depending on what are the constraints or measurements used in _your_ description of the system.
If you choose to describe the system using its microstate - and you know it - there are no probabilities anywhere.
You can of course know something and choose to ignore it - the entropy is still a reflection of the uncertainty (actual or for the sake of a lower-resolution model).
There is some objective nature to the operational definition of entropy based on an experimental setup where you fix the volume and measure the temperature or whatever.
And this is related to the statistical mechanical definition of entropy based on the value of the corresponding state variables.
But it’s not a property of the microstate - it’s a property of the macrostate which makes sense only in the context of the experimental constraints and measurements.
If we relate entropy to work that can be extracted someone with a better understanding of the state of the system and operational access to additional degrees of freedom can extract additional work.
Thermodynamics assumes the state variables provide a complete description of the system. Statistical mechanics assumes the state variables provide an incomplete description of the system - and work out what that entails.
> But it’s not a property of the microstate - it’s a property of the macrostate which makes sense only in the context of the experimental constraints and measurements.
The same can be said about the wavefunction then, right? You can't directly observe it, you can only use it to predict the statistics of a particular experimental setup. So, at worse, entropy is as real as wavefunction amplitudes.
> If we relate entropy to work that can be extracted someone with a better understanding of the state of the system and operational access to additional degrees of freedom can extract additional work.
Is this actually true? Per my understanding, if I give you three containers, two of which are filled with some kind of gas that you know nothing about, and the third with a mix of those same gases, you can measure their entropy using thermodynamic experiments and tell which of the three is a mix of the other two because it will have a higher entropy. So, you can extract more work from one of the boxes despite not knowing anything more about it.
> Per my understanding
What’s the source of that understanding? You cannot measure the entropy, only changes of entropy - which will be the same (for an ideal gas).
Edit: we already had this discussion, by the way: https://news.ycombinator.com/item?id=42434862
But the point is that, regardless of how you choose to describe or even measure the system, it will need exactly as much heat to raise its temperature by 1 degree (or it will need as much kinetic energy to increase the average velocity of the constituents by the same amount, in the microstate framework). So there is some objective nature to entropy, it's not merely a function of subjective knowledge of a system. Or, to put it another way, two observers with different amounts of information on the microstate of a system will still measure it as having the same entropy.