Maximum Entropy Principle for the Microcanonical Ensemble

We derive the microcanonical ensemble from the Maximum Entropy Principle using the phase space volume entropy of Gibbs. Maximizing (or extremizing) the entropy with respect to a general probability distribution and using the constraints of normalization and average energy, we obtain the condition that the energy is a constant E that characterizes the microcanonical ensemble. We justify the phase space volume entropy of Gibbs by showing that the combined first and second laws of thermodynamics is satified, a condition that Boltzmann called orthodicity. We also show that the entropy calculated from the Tsallis q-escort probability distribution approaches the phase space volume entropy in the limit as q approaches minus infinity. Our approach is in contrast to the commonly accepted derivation of the microcanonical ensemble from the Maximum Entropy Principle that assumes a priori that the energy E is a constant. Then the Shannon information theory entropy with only the constraint of normalization gives Laplace's Principle of Insufficient Reason for the states with the constant energy E.