Reference: authoritative reference on Gibbs and Boltzmann's entropy Can someone reference a good, standard textbook on thermodynamics or quantum mechanics that explicitly states the formula for Gibbs and Boltzmann's entropy (or maybe Shannon as well)?
I am asking because I have seen many papers where authors use these terms interrchangably and sometimes will even say Gibbs-Boltzmann entropy so I have became very confused.  
 A: Gibbs vs Boltzmann entropies
The question was discussed by Jaynes in  Gibbs vs Boltzmann entropies. American Journal of Physics, 33(5):391–398, 1965 (also availableble from here as Ref 21).
One must be careful by what we mean "Boltzmann entropy" because Boltzmann tried a lot of different things and came across various different aspects of entropy.
Boltzmann entropies

*

*Perhaps the most well known Boltzmann entropy is the equation
$$
\tag{1}
   S = k \log W
$$
which is inscribed on his tomb. Today we understand this to represent the entropy of an isolated system with fixed energy, volume, and number of particles.


*Boltzmann also came up with the expression
$$
\tag{2}
   S = -k \sum_i p_i \log p_i
$$
by discretizing the phase space of particles.


*Boltzmann obtained yet another entropy via his $H$ theorem:
$$
\tag{3}
   S = - \int f(v) \log f(v) dv
$$
where $f(v)$ is the velocity distribution of a single particle in a dilute gas. As Jaynes shows, this entropy converges to the Gibbs entropy in the ideal-gas limit.
Gibbs entropy Gibbs wrote the following equation for the entropy:
$$
\tag{4}
\boxed{
   S = - k\int \rho(\Gamma) \log \rho(\Gamma) d\Gamma
}
$$
where $\Gamma = (\mathbf r_1,\cdots;\mathbf q_1,\cdots)$ is the vector of positions and momenta of all particles and $\rho(\Gamma) d\Gamma$ is the $N$-particle probability, namely, the probability to find the microstate of the entire system in the the region $(\Gamma,\Gamma+d\Gamma)$ of phase space. This is the entropy of statistical mechanics.
Shannon entropy Shannon obtained the entropy of generic probability distribution $p_i$ as
$$
\tag{5}
   S = - \sum_i p_i \log p_i
$$

Relationship between the various entropies

*

*Gibbs's entropy is Shannon's entropy in the continuous domain applied to the equilibrium distribution of microstates.


*Boltzmann's entropy in Eq 1 is obtained from Gibbs entropy when all microstates are equally probable (the case in an isolated system, i.e., a microcanonical ensemble).


*Boltzmann's Eq 2 is analogous to the Gibbs/Shannon entropy.


*Boltzmann's entropy in Eq 3 represents thermodynamic entropy only in the ideal-gas limit (no inter-molecular forces, i.e., no correlation between states of particles).
