Legendre transform from entropy to Helmholtz free energy

Start from an entropy representation $S(U,V,N)$, such as the microcanonical entropy derived from the density of states . The temperature is obtained by differentiating the entropy with respect to energy , so the variable conjugate to $U$ is $1/T$ (or $\beta=1/(k_B T)$ from inverse temperature ).

Definition (Legendre–Fenchel transform).
The Helmholtz free energy $F(T,V,N)$ is constructed by trading $U$ for $T$ via a (Fenchel) Legendre transform :

The minimizing energy $U^\star(T,V,N)$ satisfies the stationarity condition

A common equivalent dimensionless form is

This makes explicit that the transform is “sup” rather than “inf” in the entropy representation because $S$ is typically concave in $U$ for stable macroscopic systems.

Physical meaning.
This construction produces the thermodynamic potential appropriate to the canonical ensemble : fixing $T$ means the system explores energies, and $F$ captures the competition between energetic cost $U$ and entropic gain $T S$. In the thermodynamic limit , the supremum form aligns with the Laplace principle interpretation of Laplace-type integrals.

Connection to the partition function.
For canonical statistical mechanics, the same object is obtained from the canonical partition function through free energy from the partition function :

This identification explains why derivatives of $F$ generate equilibrium response functions (e.g. pressure and chemical potential) and underlies extracting observables from $\log Z$ .