Definitions of free energy

• fes
• free_energy

In the literature, two different definitions of the free energy are common:

1. thermodynamic / probabilistic free energy
2. geometric free energy

The question is how to interprete the delta function in the following definition [¹]

F(s) = - \beta^{-1} \ln \int_{R^n} \exp(-\beta V(x)) \delta(s)

What are their differences/similarities?

Footnotes