can be found in the previous works of Katz [@1978MNRAS.183..765K; @1979MNRAS.189..817K] and Sorkin [@Sorkin:1981jc], and a more intuitive interpretation has also been given in [@1994PhLA..185...21K].

Suppose $\mathcal{M}$ is the space of possible configurations of a system in which a given configuration is specified by a point $X$. The set of independent thermodynamic variables ${\mu^i}$ specifies a given ensemble. Let $Z$ be the corresponding Massieu function. Both ${\mu^i}$ and $Z$ are functions on $\mathcal{M}$. Equilibrium states occur at points in $\mathcal{M}$ which are extrema of $Z$ under displacements $dX$ for which $d \mu^i = 0$. At equilibrium the Massieu function depends only on the values of the thermodynamic variables $\mu^i$. We can define conjugate variables $\beta_i$ such that $$dS = \beta_i d \mu^i
 \label{eq:dS}$$ for all displacements $dX$. The set of equilibria is a submanifold $\mathcal{M}_{eq}$ of the configuration space. Points in $\mathcal{M}_{eq}$ can be labeled by the corresponding values of $\mu_i$ which are called control parameters. Usually all that we know is the explicit expression of the equilibrium Massieu function $$Z_{eq} = Z (\mu^i),$$ which is the integral of Eq. (\[eq:dS\]). The explicit expressions for the conjugate variables at equilibrium are calculated as $$\beta_i (\mu^i) = \frac{\partial Z_{eq}}{\partial \mu^i}.$$ The 