Duality of Fundamental Equations Entropy Maximization versus Energy Minimization

4 DUALITY OF FUNDAMENTAL EQUATIONS ENTROPY MAXIMIZATION VERSUS ENERGY MINIMIZATION 

The preceding sections have made clear that the functional dependence of the entropy on its natural arguments U, V, N, 

However, Gibbs demonstrated the equal importance of a second fundamental equation that reveals a beautiful duality of the thermodynamic formalism the deep symmetry between entropy (5.28) and internal energy U (5.29)  

Claim If entropy S = S(U, V.) is maximized under the constraint of constant U, then the internal energy U = U(S, V.) is correspondingly minimized under the constraint of constant S, i.e., 

The proof of this important claim is sketched in Sidebar 5.3. 

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