[hexxiiiz]

In thermodynamics there are two other formulations of entropy: the Clausius one in terms of temperature and heat, and the Boltzmann one. The latter defines entropy as the log of the number of microstates a system could be in a particular macrostate.

The Shannon definition is equivalent to the Boltzmann def only in the case that the micro state consists of infinitely many identical subsystems. If there are only finitely many, for instance, the log of the quantity does not correspond to the same "-p log p".

The Clausius def can be derived from the Boltzmann one, but they are nevertheless also distinct formulations.

[jbay808]

https://en.wikipedia.org/wiki/Boltzmann%27s_entropy_formula#...

According to Wikipedia, if you start with the Gibbs entropy (which is the same as Shannon entropy), and then assume all microstate probabilities are equal (which Boltzmann does), you get the Boltzmann entropy formula. It also says Boltzmann himself used a p ln(p) formulation.

So aren't they the same, perhaps up to a constant factor?

[hexxiiiz]

If you count the number of microstates for a given macrostate you get a hyper geometric number N!/(n_1!n_2!...) The log of this is the Boltzmann entropy. However, if you consider N to be very large or infinite, you can show using the Stirling approximation that this ends up being the Gibbs/Shannon entropy in that case. So, in general, no.

[kgwgk]

In thermodynamics / statistical mechanics there is another formulation of entropy: Gibbs entropy is different from Boltzmann entropy (and equivalent to Shanon entropy in information theory).