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Physics
Classical Physics
Thermodynamics
Thermodynamic equilibrium with fixed energy/entropy
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[QUOTE="Jonk, post: 6248867, member: 668634"] Yes of course. The goal is to prove the implication Entropy maximum principle => Energy maximum principle. As a reminder, [B]Entropy maximum principle[/B] : The equilibrium value of any unconstrained internal parameter is such as to maximize the entropy for the given value of the total internal energy. [B]Energy minimum principle[/B] : The equilibrium value of any unconstrained internal parameter is such as to minimize the energy for the given value of the total entropy. Callen shows rigorously that $$\dfrac{\partial S}{\partial X}=0 \mathrm{~~and~~} \dfrac{\partial^2S}{\partial X^2}<0 \implies \dfrac{\partial U}{\partial X}=0 \mathrm{~~and~~} \dfrac{\partial^2U}{\partial X^2}>0$$ Hence a maximum of S at constant U corresponds to a minimum of U at constant S. Now it should be enough to show that a system at constant S will be at equilibrium when $$\dfrac{\partial S}{\partial X}=0 \mathrm{~~and~~} \dfrac{\partial^2S}{\partial X^2}<0$$ since it would imply that U is minimum.We supposed that S is constant (since we want to show the energy minimum principle), but U isn't constant. The entropy maximum principle requires U constant, so how to use it here? That's what I don't understand. [/QUOTE]
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Physics
Classical Physics
Thermodynamics
Thermodynamic equilibrium with fixed energy/entropy
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