What does "accessibility" mean in statistical mechanics?
Is it an equivalent concept to accessibility in mathematical control theory?
I'll provide an example: When two systems A and B interact on a subspace of their respective spaces, i.e. where they overlap in space, does accessibility of states of B from A mean the part where they overlap?
When people say "accessible microstates" it means "microstates consistent with a set of constraints or conditions which you are supposed to keep in the back of your mind".
The most common example of a constraint is a fixed amount of energy. A closed physical system with exactly one constraint, fixed total energy, is called the microcanonical ensemble.
Another type of constraint/condition could be contact with a large thermal reservoir $R$ at temperature $T$. Such a system $S$ in contact with $R$ is modeled by the canonical ensemble.
So, "accessible" just means "consistent with the constraints and conditions at hand".
"Accessible microstates" is related to things like activation energy and metastability.
For example, think about a diamond sitting on a table at room temperature and atmospheric pressure. There are microstates for the diamond in which the carbon atoms are rearranged into a piece of graphite. But those microstates will essentially never occur no matter how long you wait and stare at this diamond sitting on the table. So the system has these microstates, where the diamond's atoms are rearranged into graphite, but these microstates are not "accessible" microstates under the circumstances. We need to ignore these microstates when computing the diamond's heat capacity etc.
It's not because the diamond is more stable (lower-energy), because it's equally true the other way around: A lump of graphite on a table will not spontaneously reorganize into diamond.
(For what it's worth, I think I heard somewhere that graphite, not diamond, is the most stable form of carbon at standard temperature and pressure.)
