What are the exact conditions under which one can say for sure that the entropy of a system will increase?

In their famous paper on the second law of thermodynamics, Lieb and Yngvason state the entropy principle in the following form (paraphrased by me):

"The entropy of a system after it has transitioned to a state adiabatically accessible from some other state always either increases or stays the same."

They derive this result from some axioms and they prove that this is valid even for composite systems, in which the principle applies to the total entropy, calculated as the sum of the individual entropies.

Callen, in his famous book on thermodynamics, states, in page 133 the following:

"The equilibrium value of any uncontstrained internal paramenter is such as to maximize the entropy for the given value of the total internal energy".

Callen talks a lot in his book about extremizing thermodynamic potentials when there are constrained parameters, but he seldom mentions adiabaticity when discussing states that maximize entropy (at least as far as I remember). Callen also uses these principles when talking about composite systems and he derives this way, for instance, the conditions for thermal, mechanical and chemical equilibrium in an equilibrium state.

Callen, in turn, seems to be paraphrasing Gibbs, who, in his paper "On the equilibrium of heterogeneous substances" says something similar (in page 2):

"For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy, the variation of its entropy shall either vanish or be negative"

If I understand correctly, Lieb and Yngvason seem to be stating that entropy always increases in adiabatic processes, while Callen and Gibbs restrict themselves to isoenergetic processes. I have been told throughout my undergraduate studies that constant energy and volume (I assume that constant energy, volume and other "work coordinates" or non entropic extensive variables) are the necessary conditions for entropy to be maximized in an equilibrium state (in other words, the system should be completely closed, or isolated).

What are then the actual necessary conditions for the entropy to be maximized in an equilibrium state? What are the exact, minimal conditions under which one can be completely certain that the entropy of a system will increase between two equilibrium states?

My intention is not to point out any sort of contradiction or problem in the writings of either Callen or L & Y, it is just that this has always been a confusing topic to me and I have never been able to understand it completely satisfactory way. I hope I am not misunderstanding the writings of these authors, please point it out to me if I am.

• What about stirring of a fluid in a thermally insulated container using a paddle? The process is adiabatic but entropy increases. – Deep Oct 8 '17 at 3:15
• I see no problem with that at all, but I also don't see what it proves. – Ignacio Oct 8 '17 at 3:25
• Sorry my bad. I misread L & Y's statement to refer to only adiabatic processes. – Deep Oct 8 '17 at 3:28
• I guess I understand the idea now that you said that though, it seems to be impossible for entropy to diminish in any sort of adiabatic process, not only in an isoenergetic one, isn't that right? but why the emphasis on closed systems in Callen (and everywhere for that matter)? Is it only a very important particular case? – Ignacio Oct 8 '17 at 3:33
• If we are allowed to add energy or matter to a system, it is trivial to make its entropy decrease. I mean, consider the system formed of a 10 grams of crystallin salt and a liter of water: you put the salt into the water and let it dissolve: entropy increases. Then you evaporate the water and condense it: you can recover the initial state of 10 grams of crystallin salt and 1 liter of water, thus decreasing entropy. This is so obvious that I must have missed something in that last question of yours. – user154997 Oct 8 '17 at 16:27

• Fixed $$U$$, $$V$$, $$n_i$$: Maximize $$S$$
• Fixed $$S$$, $$V$$, $$n_i$$: Minimize $$U$$
• Fixed $$T$$, $$V$$, $$n_i$$: Minimize $$F= U - TS$$
• Fixed $$T$$, $$P$$, $$n_i$$: Minimize $$G = U - T S + P V$$