# Confusion regarding isolated systems, entropy and state space

I've been trying to understand more deeply the idea of entropy and of state functions, and become rather confused with isolated systems in the process.

Consider an isolated thermodynamical system (meaning $dW=0$, $dQ=0$, $dN=0$) in equilibrium. The system "occupies" a specific point A in the space of macrostates. A reversible process taking the system to point B is represented by a continuous curve in said space joining both points. Any two points can be joined at least in principle by such a curve, but as it would represent a reversible process, the conclusion would be that $S(A)=S(B)$, i.e. $S$ is constant over all state space for an isolated system in equilibrium.

I don't really think this reasoning is correct but I'm not sure why. It could be argued that in order to devise an arbitrary reversible path between A and B, one would need access to (for instance) an infinite number of heat reservoirs exchanging heat with the system in a convenient way, and that contradicts the isolation assumption. If it were actually correct, I would make sense of it this way: an isolated system either starts out of equilibrium (meaning it starts "off" state space) and finally settles at one specific point of state space, or starts in equilibrium, thus staying at the same point indefinitely. However, as far as I understand, this would imply that an isolated system in equilibrium can undergo reversible processes but not irreversible ones, because the latter would imply an increase in entropy and "there's nowhere to go" in state space to fulfill such requirement. Finally, the picture gets even more confusing if I add the possibility of changing constraints. The entropy of the system might change after stabilising on a new equilibrium after a constraint is added or removed, but if the function $S$ was constant over all state space, then the function $S$ itself would have changed! What's really going on here?

I've realised I don't even know if "isolated system" means its constraints can't be modified either. For instance, there is no work nor heat exchange involved in the expansion of the available volume for an ideal gas (as in moving a piston externally). If an isolated system's constraints can be modified, then why can't there exist some weird system for which the maximum value of entropy consistent with the constraints decreases when changing its constraints in a convenient, zero-work zero-heat fashion? Then the Second Principle would be violated, so something must be wrong. I guess either the definition of "isolated" includes fixing the constraints, or such particular system can't exist for some reason I'm not considering, or maybe changing the constraints simply changes the definition of the system.

• For your infinite number of heat reservoir example, the reservoirs would have to be included as part of the isolated system. In this way, the change in entropy of the combined system would be zero for the reversible path. – Chet Miller Sep 5 '16 at 15:01
• The reservoirs could be included in the isolated system, of course, but I don't see how that would be helpful. The main reasoning I presented requires an isolated system starting at equilibrium, with well-defined macrostate parameters (implying a unique temperature). Wouldn't what you say prevent us from defining the initial temperature? Or maybe not and I should go over the idea of composite systems; that might be part of my confusion... – GBRGR Sep 5 '16 at 15:23
• I agree. You should consider the idea of composite systems. Composite systems don't need to be all at the same inital temperature. They are constrained by being apart, until you allow them to come together in sequence so that no parts of the composite system are more than differentially removed from thermodynamic equilibrium during the entire change. – Chet Miller Sep 5 '16 at 15:55