# Is Entropy a monotonically increasing function of Gibbs Free Energy/ Helmholtz free energy/ Enthalpy?

Entropy can be axiomatically taken as a monotonically increasing function of internal energy $$(E) ,$$ from where "energy minimum principle" can be deduced, and this can be stated using variational principle in mathematical language. (Ref: Introduction to modern statistical mechanics, by David Chandler, Chapter 1)

Similar variational principles can be derived for Gibbs free energy $$(G) ,$$ Helmholtz free energy $$(A) ,$$ and enthalpy $$(H) .$$

Question: Do similar reasonings (or axioms) exist for entropy which are related to Gibbs free energy, Helmholtz free energy, or enthalpy?

• "Entropy can be axiomatically taken as a monotonically increasing function of internal energy" - This isn't true in certain quantum systems. For example, certain negative-temperature systems have a well-defined highest energy state; as the internal energy is increased, more and more particles are packed into the same state, and the entropy decreases. Also, be careful about the definition of "monotonically increasing," as there are differing conventions about whether a monotonically-increasing function can stay constant over some interval. – probably_someone Jan 22 at 22:41
• @Nat "esthetic" is a valid alternative spelling for "aesthetic." See e.g. merriam-webster.com/dictionary/aesthetic – probably_someone Jan 22 at 22:43
• @Nat I understand that the aesthetic changes are less important, but it is a practice that according to me makes it easier to differentiate between text and variables like "Gibbs free energy ($G$)". Besides, it makes reading a little more rewarding, I think. – El borito Jan 23 at 0:20
• @probably_someone, Thanks you for your indicating points. "Monotonically increasing" may create confusion indeed. But the text does not give any clue about what convention it is following.I guess there is no region where the function stays constant, according to the text and its following calculations. And hopefully this part of the text starts with classical thermodynamics where quantum consideration is not taken into account.By the way, would you kindly elaborate your example? What does "negative temperature" mean?And were are talking about stable equilibrium where the entropy is maximum. – Deepayan Turja Jan 27 at 9:26