When we speak about minimization of free energy there are two approaches, one from statistical physics and one from thermodynamics:
- Statistical physics tells that is we consider a system in thermal contact with a heat reservoir, so that $T_r, V$ and $N$ are fixed parameters, the free energy is $F=-k_{\mathrm{B}}\,T\ln(Z)$. Then, consider an internal fluctuating variable $Y$, the free energy of the system with $y$ fixed to the value $y_0$ is $F(y_0)=-k_{\mathrm{B}}\,T\ln(Z(y_0))$. It is possible to show that the thermodynamic value associated with the fluctuating variable y will be the one such that $\frac {\partial F(y)} {\partial y}=0$.
- Thermodynamics teach that for a system at constant $N,V,T$ the thermodynamic equilibrium is when $F$ is minimized.
I see a strong analogy between these two approach. I suspect that the first one is just the statistical physics justification for the second however I'm not sure because $F(y)$ is not the total free energy of the system and thermodynamic says that the total energy of the system is minimized. Can you give help me in giving an order to these, maybe analogous approach to free energy minimization?
