From a purely thermodynamical point of view, why does that entropy have to be a maximum at equilibrium? Say there is equilibrium, i.e. no net heat flow, why can the entropy not be sitting at a non-maximal value?
From the second law of thermodynamics, it follows that $S$ never gets smaller
and of course I know that for an isolated system there are many statements involving $\text{d}S=\frac{\delta Q}{T}$, which say what might happen for processes. But if I have equilibrium, then no relevant processes are going on. Some proofs in thermodynamics involve arguments how if we don't have maximal entropy, then we can do something which raises entropy. But why is that relevant or related equilibrium, i.e. to the positions for the termodynamic parameters, which don't change with time? One could argue that probably the energy $U$ doesn't sit at the minimal value, but in thermodynamics, without microscopic forces, the statement that the energy changes towards equilibrium and seeks its minimal value seems to be derived from maximal entropy.
And how can I conclude the converse? Why does equilibrium follow from $\text{d}S=0$?
(edit: I see the question was just bumped on the front page, and as it's a year old now, I guess my interest in the topic has changed in so far as I'm not particularly happy with the formulation of the initial question anymore. That is, I guess without proper stating the definitions, it might be difficult to give a good answer – the thing I still don't "like", and that might be kind of a language problem, is that "to maximize" implies that there is a family of values it could take – but it takes the maximum – while at the same time, you often consider a deviation away from a thermodynamical state to be a transition into a configuration where thermodynamics doesn't apply anymore. Hence, if you go away from the "maximal value", you might lose the concept of there being a temperature at all, but since this is what makes the parameter space with respect you use the word "maximal" entropy, you get into language problems. But at least I do see it's use in explainaing how the entropy/energy evolves once you take the extensive parameters into your control and change the system quasistatically. In any case, I am and was only only interested in a non-statistical mechanics answer here. Clearly, I can make sense of the physics using the microscopic picture anyway, but was purely interested in the formulation of the thermodynamical theory here.)
Best Answer
First, if ${\rm d}S\neq 0$, then the entropy will change, and because something is changing, it's obviously not an equilibrium.
If the physical system doesn't maximize the entropy and it's composed of many parts that may interact with each other, directly or indirectly (it's not disconnected), then any path that allows the entropy to be increased (given fixed values of conserved quantities such as energy) will be realized, so you will be away from the equilibrium because something will change.
If a system is composed of two or more decoupled, non-interacting components – like a bottle of blue lemonade and a bottle of red lemonade – they may be in equilibrium even if the entropy isn't maximized. One could increase the entropy by mixing the liquids but because they're not in contact, they won't be mixed.
On the contrary, if the entropy is already maximized, the only way how the system may evolve is to evolve into another state with the same, maximum value of entropy: there's no higher allowed value and the second law of thermodynamics prohibits a decreasing entropy. This is atypical because when we maximize entropy among all states with the same conserved quantities, the state of maximum entropy is typically unique. For example, if there are also movable macroscopic bodies that may create heat by friction, the entropy is maximized only when the friction stops the macroscopic motion and converts its energy to heat.
In all these discussions, one has to be careful on whether or not we're maximizing the entropy among all states or just the states with the same value of energy (and other conserved quantities). If we allow the energy to change arbitrarily, the entropy isn't really bounded from above (and discussions about its maximization are rendered meaningless) because any body may be heated to pretty much arbitrarily high temperature (or it may collapse into a black hole with an ever greater mass and therefore an ever greater entropy, too).