Sketch of statistical mechanics

Consider N fermions in a box with fixed energy levels E, (including rest-mass which itself includes internal excitation energy). To each E, there correspond co, distinct (degenerate) states, made up of gi internal states and 4n Vp2dp/h3 kinetic degrees of freedom, where p is momentum and V is the volume. 

Because fermions are indistinguishable and satisfy the Pauli Exclusion Principle, the number of different ways in which each set of oj, states with energy E, can have N, of them occupied is 

Thermal equilibrium implies that, under fixed conditions, W takes on its maximum possible value, i.e. 

These conditions are incorporated into the stationary condition (2.29) using Lagrange multipliers 

Considering this thought-experiment from the point of view of thermodynamics, the additional particles add enthalpy, 

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