Fundamental relation (entropy representation)

In the entropy representation, the fundamental relation is the equilibrium entropy expressed as a function of the extensive variables: $S = S(U,V,N,\dots)$, where $U$ is internal energy , $V$ is volume , and $N$ is particle number (and possibly other conserved extensive quantities). This single state function determines the equations of state and all response functions for systems in thermodynamic equilibrium .

Physical interpretation. The function $S(U,V,N,\dots)$ counts (macroscopically) how many microscopic configurations are compatible with macroscopic constraints; its derivatives define the intensive “forces” that drive exchange with the surroundings .

Generating derivatives. The differential form of the fundamental relation is

which defines the intensive variables as partial derivatives:

• $\frac{1}{T} = \left(\frac{\partial S}{\partial U}\right)_{V,N,\dots}$ with temperature
• $\frac{P}{T} = \left(\frac{\partial S}{\partial V}\right)_{U,N,\dots}$ with pressure
• $-\frac{\mu}{T} = \left(\frac{\partial S}{\partial N}\right)_{U,V,\dots}$ with chemical potential

From these, one can obtain an equation of state such as $P=P(T,V,N)$ by eliminating $U$ in favor of $(T,V,N)$ using the derivative relation for $T$.

Key properties.

• Extensivity as homogeneity. For additive systems away from long-range interaction effects, the entropy is extensive and (ideally) homogeneous of degree one in $(U,V,N,\dots)$, which underlies the Euler relation and Gibbs–Duhem relation .
• Concavity and stability. Stability in the entropy representation corresponds to concavity of $S$ in its extensive arguments, linking to entropy concavity (stability) and constraining response functions (e.g., positive heat capacities in typical regimes).
• Second-law compatibility. The entropy fundamental relation is consistent with the second law and the Clausius inequality for irreversible processes.