What is the Gibbs free energy difference between conformational states In statistical mechanics, for a system of $N$ particles $x_1, \ldots, x_N$ in three dimensions, the Gibbs free energy is defined in terms of the Hamiltonian $H$ as
$$ G = -k_\mathrm B T \log \int_{\mathbb{R}^{3N}} \mathrm e^{-H(x_1, \dots, x_N)}\,\mathrm dx_1 \dots \mathrm dx_N.$$
People talk about the "free energy difference between conformations". For instance, for some region $A \subset \mathbb{R}^N$, the particles could be in a certain conformation, and for some other region $B \subset \mathbb{R}^N$, the particles could be in another conformation.  

Question


  In this paper, equation (3), what is the (implied) definition of $F_A$ or $F_B$ in terms of equation (2)?


Equivalent Question


  Is the free energy difference between the two conformations defined as
  $$G_B - G_A = -k_\mathrm B T\log \frac{\int_B \mathrm e^{-H(x)}\,\mathrm dx}{\int_A \mathrm e^{-H(x)}\,\mathrm dx}?$$


Auxiliary details

The above definition made sense to me, until I was told that "it does not make sense to talk about the free energy of a conformation". An alternative I see is to think of two Hamiltonians $H_A$ and $H_B$, and instead define the free energy difference as

$$G_B - G_A = -k_\mathrm B T \log \frac{\int_{\mathbb{R}^{3N}} \mathrm e^{-H_B(x)}\,\mathrm  dx}{\int_{\mathbb{R}^{3N}}\mathrm e^{-H_A(x)}\,\mathrm dx}. $$

Are either/both/neither of these definitions correct?

I apologize if the question is exceedingly obvious. I am unable to find a straightforward answer by Googling. Any references are welcome.
Your question implies that you define the Gibbs free energy as $G=-RT\ln(Z)$ where $Z$ is the partition function, in which case there will be a difference in $G$ only if there is a difference in the partition function for your two configurations, i.e differences in energy levels and/or in the population in these levels. That will depend entirely on your system (and also the temperature).