I have a question about a characterization of thermodynamic equilibrium as given on German wiki article: https://de.wikipedia.org/wiki/Thermodynamisches_Gleichgewicht#Abgeschlossenes_System
What does it state? Let $S=S(U, V, N)$ the entropy and the expression $S=S(U, V, N)$ make sense as well a state with macroscopical parameters $U,V,N$ is in thermodynamical equilibrium. The claim is
"Ein abgeschlossenes System befindet sich im thermodynamischen Gleichgewicht, wenn seine Entropie $S$ maximal ist. Entsprechend gilt, für das Differential
Translated that means
A closed system is in thermodynamic equilibrium when its entropy $S$ is at a maximum. For the differential this means $dS=0$.
And the last condition I not understand. By definition writing $S= S(U,V,N)$ (or for any other arbitrary thermodynamic potential e.g. internal energy $U(S,V,N)$ or free Helmholtz $F(T,V,N)$)
makes only sense if the given state parametrized by macro parameters $U,V,N$ is already in thermodynamic equilibrium. Otherwise, $S= S(U,V,N)$ don't make any sense as a thermodynamical system that is not in a thermodynamic equilibrium is too complex to be described by only three independent macro parameters $U, V,N$ and futhermore the concept of associating a thermodynamical potential to a state only works if one consider a state beeing already in equilibrium.
Therefore what is $dS=0$ (or more precisely how to interpret it in this context?) and why does it make sense to use it as characterization of equilibrium?
The only way I'm familar how the differential $dS$ is used in thermodynamics is explaned in following setting. We start with a state parametrized by $(U_1,V_1,N_1)$ that is already in thermodynamic equilibrium and then we start a certain thermodynamical process (reversible or irreversible) that finally brings the system in another state $(U_2,V_2,N_2)$ that is also in a new thermodynamical equilibrium after a long time.
The point is that how we pass from $(U_1,V_1,N_1)$ to $(U_2,V_2,N_2)$ we don't know precisely as in nature we pass during the process non equilibrium states that we can't describe with our formalism. One possible ansatz is to consider it a sequence of quastatical processes such that each intermediate state is assumed to be also in thermodynamical equilibrium. This is of course a strong idealisation.
With this idealization passing from $(U_1,V_1,N_1)$ to $(U_2,V_2,N_2)$ can be indeed visualized as a curve in Phase space as long as we consider it as a quasistatic process.
then indeed also the expression $dS= S(U_2,V_2,N_2)-S(U_1,V_1,N_1) $ make sense.
But in this case it makes no sense to use $dS=0$ as characterization of a state beeing in themodynamic equilibrium as by $dS$ we consider allways diferences of enropies of states that are already in equilibriums.
Does anybody have an idea how the "characterization thermodynamic of quilibrium" $dS=0$ should here be understood and what is the error in my reasonings above?