I recall reading in Negele&Orland's book that Landau free energy function is not really the free energy that one obtains from the partition function $$F = -\frac{1}{\beta}\log Z.$$ Indeed, the real free energy is already averaged over all possible states and its expansion will never produce a nice shape with multiple minima, used in Landau's analysis of phase transitions.

Is Landau energy just a useful theoretical conjecture or is there a principled way of constructing it from a partition function? Examples will be appreciated.


Alternative definitions of free energy:
Microcanonical ensemble
Given the density-of-states $\Omega(E)$, we have (taking $k_B=1$) $$ S(E) = \log \Omega(E),\\ \frac{1}{T}=\frac{\partial S(E)}{\partial E},\\ F(E)=E-TS(E) $$

Canonical ensemble
$$ Z=\sum_n e^{-\frac{E_n}{T}}=\int dE \Omega(E)e^{-\frac{E}{T}}= \int dE e^{-\frac{E-T\log\Omega(E)}{T}}=\\\int dE e^{-\frac{F(E)}{T}}=e^{-\frac{F_0}{T}} $$ When taking the thermodynamic limit, the integral can be evaluated using the steepest descent method, which gives $$ F_0=\min_{E}F(E) $$

In other words, the free energy in canonical ensemble (which is meant by Negele&Orland cited in the OP) is actually the minimum free energy in terms of that of the microcanonical ensemble. Calling Landau function free energy certainly implies the microcanonical definition, which si used in many physical/chemical methods aiming at free energy minimization.


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