Problem understanding entropy (two different definitions?)

[James Brown]

In the definition of entropy, there are two. One is about the degree of randomness and one is about energy that is not available to do work. What is the relationship between them?

 

[TeethWhitener]

There are several definitions of entropy. But let's say you're asking about the Gibbs (statistical) entropy vs. the classical definition of entropy. The box on the right side of this wiki page:
https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)#Gibbs_entropy_formula
gives a straightforward demonstration of the equivalence of the Gibbs entropy formula and the classical entropy formula assuming a canonical ensemble.

 

[Demystifier]

In the definition of entropy, there are two. One is about the degree of randomness and one is about energy that is not available to do work. What is the relationship between them?

The former is statistical entropy and the latter is thermodynamic entropy. Statistical entropy is more general, while thermodynamic entropy is only defined for systems close to thermodynamic equilibrium. It can be shown (see the box in the link in the post above) that statistical entropy for systems close to thermodynamic equilibrium can be well approximated by thermodynamic entropy.