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

Question

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.

Answer 1

To talk about stability we must first agree on the variables that define the state of the system. The specification "adiabatic" system is not sufficient. Here are some examples:

• 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 $G = U - T S + P V$

In all cases the exremization is with respect to partitioning of the extensive variables while holding all intensive variables constant.

Answer 2

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.

You are comparing different trains of thought, which are consistent but different, so there will be some differences, this should not cause confusion.

L&Y work with the concept of adiabatic accessibility and finite processes between different macroscopic states: and claim that if system can reach equilibrium state X2 from equilibrium state X1 by a process which may involve work but must not involve heat transfer at any point, then entropy change (S2-S1) can't be negative. It does not follow entropy has to increase. It may stay the same. In general, these processes may be violent (explosion, shock waves etc), they need not be quasistatic. A special case of this are quasistatic changes, where X2 is connected to X1 by a path in the space of equilibrium states.

Callen and Gibbs quotes are about property of equilibrium states, not about real processes. The processes they implicitly consider when stating the equilibrium condition are virtual processes, or infinitesimal processes, that tweak values of state variables a little around their values for the equilibrium state X, while keeping energy constant. These processes are purely hypothetical and it does not matter in this context whether they are adiabatic or not. What matters is that they must be quasistatic, that all thermodynamic variables are defined at every stage of the process. For example, if a system is in equilibrium state X, and if a wall can be moved a little and a small heat accepted while its energy stays unchanged (in the first approximation), this is a virtual process in which system's entropy change in the first approximation is zero (because in equilibrium it is at its maximum) and in second approximation it may be zero or negative (if the equilibrium is stable, then it is negative).

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?

There are two different questions!

1. Entropy is always maximized(more correctly: at a stationary point) in any equilibrium state. It may not be a maximum but certainly first derivatives of S are zero. This maximum/stationary point is as compared to various nearby states with the same energy and volume (and possibly other external constraints such magnetic field) but different internal details of subsystems (such as values of positions of internal walls or particle numbers for different phases/partitions)

2. There is no simple general rule to detect all processes that increase entropy. A sure way to increase entropy is to keep heat coming into the system from the outside but forbid any outgoing heat. But this does not capture all processes that increase entropy. Heating can be simulated by doing dissipative work. One can oscillate a piston very quickly or stir the fluid inside a vessel and then entropy will also certainly increase, because the work effect is equivalent to some heat. Also, if the system has some internal constraints such as walls or activation energy for chemical reaction, then a sure way to increase its entropy is to remove that wall/start the reaction; the spontaneous process that ensues will result in a new state of higher entropy.

There is only a general rule to detect processes that can't decrease entropy; the pure work processes. Such processes will keep entropy constant (if quasistatic, which is almost impossible in reality) or will increase entropy (all processes, the faster / more violent the process, the more entropy is generated).