Evolution of a system - Thermodynamic potential There is something I don't understand about thermodynamic potentials.
We know that the necessary condition for a system to evolve is an increase in entropy. For a system kept at constant volume and temperature for example, this means :
$$dU = \delta Q, \hspace{7mm} dS>\frac{\delta Q}{T} = \frac{dU}{T}$$
$$\Leftrightarrow TdS > dU$$
$$\hspace{2cm}\Leftrightarrow d(U-TS) = dF < 0$$
But if $V$ and $T$ are held constant, $dV=dT=0$.
So how is $\hspace{3mm}dF = -SdT - pdV \hspace{3mm}$ supposed to be anything else than zero ?
 A: Your thermodynamic system has 2 degrees of freedom. We can place all states of the system on a 2d plane, which we can parametrise by whichever 2 thermodynamic variable we choose, lets say $S$ and $V$ for definiteness.
If we impose a constraint upon the system, such as fixing the temperature, we reduce the number of degrees of freedom by 1 and the allowed states of the system now lie on a 1d curve; an isotherm in the previous example. 
If we impose another constrain, such as fixing the volume, we reduce the number of degrees of freedom again, so our system is now limited to a point; that is there is only one state of the system with that volume and temperature. If you now try and force the system to change state, you find that the system is overconstrained and you encounter a problem. You cannot keep the temperature fixed and keep the volume fixed and change the entropy.   
A: There must be unconstrained variables in addition to $T$ and $V$.
Suppose that the system consists of multiple species that can undergo a chemical reaction. At a fixed $T$ and $V$, the number of each species changes (via the chemical reaction) in such a way that $dF<0$.
