Thermodynamic Equivalence and Ensemble Choice

Explain the concept of thermodynamic equivalence of ensembles in the thermodynamic limit (N → ∞).

Show that relative fluctuations of extensive quantities (e.g., energy) vanish in the thermodynamic limit, scaling as 1/√N.

Apply the large deviation principle to show how the canonical and grand canonical potentials emerge as rate functions governing macroscopic fluctuations away from the microcanonical state.

Derive the Sackur-Tetrode equation and explain how the factor of h^{3N} resolves the Gibbs paradox of mixing entropy.

Justify the choice of a specific ensemble (microcanonical, canonical, or grand canonical) for solving a given physical problem based on constraints and computational convenience.