On Wikipedia for Microcanonical Ensemble it says,
$ \omega $ is said to be an energy width. I am not entirely sure what that means, but I believe it means that it is a variable stuck in to annihilate the units (like $ h $).
I am only familiar with the Boltzmann Entropy, but by further reading, it seems that only the volume entropy reproduces thermodynamics.
The volume entropy Sv and associated Tv form a close analogy to thermodynamic entropy and temperature. It is possible to show exactly that
$ dE=T_{\rm {v}}dS_{\rm {v}}-\langle P\rangle dV, $
Ignoring the warning that preferred solution to these problems is avoid use of the microcanonical ensemble
, I am concerned at where these entropies come from. I have a reasonable intuition for the Boltzmann entropy as is is a special case of entropy from information theory, but these other entropies seem to not be based on information theory. Furthermore, I have trouble finding additional material on them.
Can you explain the origin of volume entropy and surface entropy? If possible explain their origin from information theory so I can understand them better.