Fundamental relation (energy representation)

In the energy representation, the fundamental relation is the equilibrium internal energy written as a function of the extensive variables: $U = U(S,V,N,\dots)$, with $S$ the thermodynamic entropy , $V$ the volume , and $N$ the particle number (plus any other extensive conserved quantities). Knowing this single state function determines all equilibrium thermodynamics for a system in thermodynamic equilibrium .

Physical interpretation. The function $U(S,V,N,\dots)$ tells you the energy cost of “building” a macrostate with given entropy, size, and composition. Its derivatives are the intensive variables that govern exchange of energy, volume, and particles across the system boundary .

Generating derivatives. The differential form is

which defines

• $T=\left(\frac{\partial U}{\partial S}\right)_{V,N,\dots}$ (temperature )
• $-P=\left(\frac{\partial U}{\partial V}\right)_{S,N,\dots}$ (pressure )
• $\mu=\left(\frac{\partial U}{\partial N}\right)_{S,V,\dots}$ (chemical potential )

An equation of state can be obtained by re-expressing these derivatives in terms of $(T,V,N)$ or other convenient independent variables.

Key properties.

• Extensivity as homogeneity. For additive macroscopic systems, $U$ is extensive and ideally homogeneous of degree one in $(S,V,N,\dots)$, implying the Euler relation and leading to the Gibbs–Duhem relation among intensive variables.
• Convexity and stability. Stability in the energy representation corresponds to appropriate convexity properties of $U$ (as a function of extensive variables), connecting to energy convexity (stability) and constraining measurable response functions .
• Link to work/heat differentials. Along a thermodynamic process , the first law relates changes in $U$ to heat and work via the first law .