Thermodynamic potential

Definition

Consider a simple thermodynamic system with internal energy $U$ given as a function $U(S, V, N_1, \dots, N_r)$ of its entropy $S$, volume $V$ and amounts of substance $N_1, \dots, N_r$. Write $x_0 = S$, $x_1 = V$ and $x_{i+1} = N_i$, and let $y_j = \partial U / \partial x_j$ be the conjugate variable of $x_j$; thus $y_0 = T$, $y_1 = -p$ and $y_{i+1} = \mu_i$. Here $T$ is the temperature, $p$ the pressure and $\mu_i$ the chemical potential of substance $i$. Let $J$ be a nonempty proper subset of $\set{0, 1, \dots, r+1}$, and consider the state function $$U[J] = U - \sum_{j \in J} x_j \, y_j.$$ Write it as a function of the variables $y_j$ for $j \in J$ and $x_j$ for $j \notin J$. A thermodynamic potential of the system is $U$ itself or such a function $U[J]$. These variables are the natural variables of the potential.