The Generalized Boltzmann Distribution is the Only Distribution in Which the Gibbs-Shannon Entropy Equals the Thermodynamic Entropy

We show that the generalized Boltzmann distribution is the only distribution for which the Gibbs-Shannon entropy equals the thermodynamic entropy. This result means that the thermodynamic entropy and the Gibbs-Shannon entropy are not generally equal, but rather than the equality holds only in the special case where a system is in equilibrium with a reservoir.