This macrostate seems fine enough to be a microstate, but sure. For the macrostates with at least one 'x' in it, that 'x' seems to be a placeholder for the concept of subjective ignorance.
> That doesn't mean I necessarily have any information about which macro-state the system is actually in.
But the entire purpose of the exercise of assigning microstates to macrostates is so that you can match up a description of some system to a microstate ensemble and calculate its entropy! Otherwise there's no point to arbitrarily labelling various groups of microstates.
To follow your example more practically, let's say you have an 8-spin system, whose net spin is zero. (You know because you've measured its overall magnetic moment or something). I've just described a system that is in one of the following possible microstates:
TTTT HHHH, TTTH HHHT, ..., HHHH TTTT
Now you can go ahead and define the macrostates as fine-grained as you want, where TTTT HHHH and HHHH TTTT are in different macrostates, but to calculate the entropy of this system, you're going to have to sum up all of those macrostates anyway to get the one that's consistent with the described system.
Good review of common ground. At this point hopefully the active folks in the discussion can see that we're all describing statistical mechanics exactly the same way.
What I'm saying is pedagogical. We need to define our macro-states. We don't need to go on and talk about our definitions being information or 'knowledge.' We could just use the definitions and calculate. We can talk about 'knowledge' but we don't need to.
The exception is when we actually have some information about what macro-state some system is really in. Obviously we then have to build that information into our model, and the entropy changes. What I'm saying is this: it's not necessary to mix that into our definition of entropy. That definition is not going to help folks who don't understand entropy, and it's unnecessary.
This macrostate seems fine enough to be a microstate, but sure. For the macrostates with at least one 'x' in it, that 'x' seems to be a placeholder for the concept of subjective ignorance.
> That doesn't mean I necessarily have any information about which macro-state the system is actually in.
But the entire purpose of the exercise of assigning microstates to macrostates is so that you can match up a description of some system to a microstate ensemble and calculate its entropy! Otherwise there's no point to arbitrarily labelling various groups of microstates.
To follow your example more practically, let's say you have an 8-spin system, whose net spin is zero. (You know because you've measured its overall magnetic moment or something). I've just described a system that is in one of the following possible microstates:
TTTT HHHH, TTTH HHHT, ..., HHHH TTTT
Now you can go ahead and define the macrostates as fine-grained as you want, where TTTT HHHH and HHHH TTTT are in different macrostates, but to calculate the entropy of this system, you're going to have to sum up all of those macrostates anyway to get the one that's consistent with the described system.