Conflicting Perspectives on Entropy and the Second Law of Thermodynamics
In classical thermodynamics, entropy is postulated to exist and be a monotone, convex, extensive state function $S(E,V,N)$.
In statistical mechanics, the picture becomes blurrier. Let's restrict our discussion to the classical statistical mechanics of isolated systems for the moment. In the words of <PERSON>, in statistical mechanics entropy becomes a function of the system and your knowledge of it. For an isolated system, $S = \log \mathcal{W}$, where $\mathcal{W}$ is the number of microstates the system can be in given your state of knowledge of the system. Assuming one knows $E, V, N$ and the Hamiltonian of the system, $\mathcal{W}$ can be computed as the volume of a particular hypersurface (or a thin shell) in the phase space.
The natural question is whether or not the second law of thermodynamics can then be proven in the context of classical statistical mechanics. Two separate accepted and upvoted answers on this site seem to provide different answers. One argues quite persuasively that you cannot, that in order to derive the second law, one needs either asymmetric laws of motion or asymmetric initial conditions. The second says the second law may be proven, citing <PERSON>'s H theorem.
I think underlying my confusion of this topics underlies a deeper confusion I have about what entropy actually means in statistical mechanics. Sometimes entropy has an almost Bayesian probability nature about it, a sort of "entropy measures one's ignorance of a system" (view 3 below.) Other contexts, entropy seems to play more like a state variable, a mathematical function $S(E, V, N)$ intrinsic to a system (views 1 and 2 below.)
1. The entropy of a system in equilibrium is $S := \log \mathcal{W}(E,V,N)$. The entropy of a system out of equillibrium is undefined.
Under this view, how can we say the entropy of the universe is increasing: the universe is not equilibrium so its entropy is undefined.
1. The entropy of a system in equilibrium is $S := \log \mathcal{W}(E,V,N)$. A system out of equillibrium is composed of many smaller subsystems each of which can be considered approximately in equilibrium. 749
The entropy of the system is the sum of the entropies of the subsystems.
This raises the question of whether the entropy of the system is independent of how it is partitioned into subsystems or how "in equilibrium" said subsystems are required to be. If these questions are resolved, the second law in this context would be "the entropy of a collection of noninteracting subsystems increases as they are allowed to interact." This is an interpretation of the second law often "proven" in elementary statistical mechanics textbooks as a hand-wavey justification of the second law. (e.g. two ideal gases of different densities separated by a wall, the wall is taken down and the entropy increases.)
In this context, the deeper "reason" for entropy increase is that the system is approaching equilibrium. But why does the system approach equilibrium and not escape from it. <PERSON>'s paradox and the assumption that the laws of motion are reversible would seem to imply it equally likely that a system would evolve away from an equillibrium state than toward one.
1. Entropy, in a strict sense, does not exist. If observers A and B both observe a system, but A observes more about the system then B, then A's computed value of entropy will be lower than B's since there are fewer microstates consistent with A's more detailed observations of the system than are micro states consistent with B's less detailed observations.
Under this interpretation, how can entropy increase be formulated at all. What if a hypothetical observer C performs more and more observations as time progresses. Won't then the number of microstates shrink as time progresses as C's knowledge of the system increases. Further, if D knows the Hamiltonian and $E,V,N$ then in principle if D computed $\mathcal{W}$ at some time, then <PERSON>'s should imply a "conservation of entropy" since <PERSON>'s theorem implies a "conservation of phase space volume". The objection, I imagine, to D's approach would be that the system originally is not in equilibrium but this just brings us back to item 1.
The more statistical mechanics I read the more confused I become about the meaning of entropy and why the second law of thermodynamics is true. Can anyone shed any insight on which, if any, of my perspectives on entropy is "correct" and how some of the logical contradictions I see can be resolved? 273