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Callen (postulate II) postulates the existence of the entropy as the quantity to be minimized (or maximized, he will talk about that later) to achieve equilibrium. Why does he postulate the entropy rather than sticking with energy? It seems like he's making a leap that he doesn't justify, even though I know from the Second Law (which he dispenses with in his presentation) that the entropy exists.

Qmechanic
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    The problem with considering energy only is that (1) every particle has a different energy, and the mean or total energy is woefully inadequate to explain how the particles evolve. (2) The sensitivity of this energy distribution width to increasing system energy is also unknown but important. From this we decide to consider the entropy and temperature. – Chemomechanics Nov 04 '21 at 04:05
  • +1. Thanks. Could you please elaborate on your statement that "the mean or total energy is woefully inadequate to explain how the particles evolve"? –  Nov 04 '21 at 04:24
  • An energy balance tells you what could occur but not what does occur. If we didn't investigate entropy, we'd predict that a cooler object could spontaneously heat a warmer object—after all, this satisfies conservation of energy. But we never see this in reality. – Chemomechanics Nov 04 '21 at 19:00
  • Thanks. So the reason we can define the equilibrium for, say, two bound atoms by the minimum in their energy well is because there is no heat flux, which requires that we also take into account the entropy as you explained? –  Nov 04 '21 at 20:02
  • Sorry, I don't know what you're referring to. – Chemomechanics Nov 04 '21 at 22:10
  • Another way of asking it -- say, you have a Hookean spring. The equilibrium position is determined solely by the minimum in the potential energy, right? So, as I mentioned, the reason we can define the equilibrium in that case by the minimum in the energy well is because there is no heat flux, which requires that we also take into account the entropy as you explained? –  Nov 05 '21 at 17:10
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    You don't have potential energy minimization without entropy and the Second Law. – Chemomechanics Nov 05 '21 at 17:24
  • +1 Thank you for the link. –  Nov 05 '21 at 17:44

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In classic thermodynamics, the existence of entropy, and that is is maximum at equilibrium, is a postulate. An axiom. This means that it is not "justified". There is no explanation of why. As you say, it is leap. Fortunately, the postulates are the only leaps. Everything else in the theory can be mathematically proven from those postulates.

The theory, with this postulate, ends up correctly predicting multiple phenomena. This is how postulates are ultimately "justified".

Note that there is another theory, Statistical Mechanics, were entropy is not a postulate, and it can be deduced from SM's own postulates. It provides a satisfying explanation of from where entropy comes.

Juan Perez
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