I am looking at some mathematics of HelmHoltz free energy. Naturally, wikipedia is one of the (hopefully) more reliable sources of information. In its derivation section, the last equation states that
\begin{equation} dA=-SdT-pdV \end{equation}
Then, in the later section, someone pointed out the apparent contradiction:
"...This result (the inequality $\Delta A \leq 0$) seems to contradict the equation $dA = -S dT - P dV$, as keeping T and V constant seems to imply $dA = 0$ and hence $A = \text{ constant}$. In reality there is no contradiction. After the spontaneous change, the system, as described by thermodynamics, is a different system with a different free energy function than it was before the spontaneous change."
I am in no position to judge the accuracy of the wikipedia text, but I can understand the apparent contradiction that is pointed out.
However, I am not satisfied with the so-called explanation that immediately follows it. My question is, how can $A$ have "different" functions of state? If we are talking about the same system, isn't it that there should be just one function for $A$ ? If so, how do we resolve the apparent contradiction that was being pointed out?