Thermodynamic-limit state function

Finite systems often exhibit size- and boundary-dependent corrections: a thermodynamic potential may depend weakly on container shape, boundary conditions, or $1/V$ effects even at equilibrium. In the thermodynamic limit , many of these corrections disappear, and properly scaled quantities become genuine state functions of the macroscopic thermodynamic state .

Definition. A quantity $A$ (or a density derived from it) is a thermodynamic-limit state function if there is a scaling (typically “per volume” or “per particle”) such that along a thermodynamic-limit sequence the limit exists and depends only on the limiting state variables (densities and/or intensive parameters), not on the path by which the limit is taken or on boundary details.

Concrete examples include:

• Helmholtz free energy density. With $F(T,V,N)$ the Helmholtz free energy in the canonical setting, one defines

  $$f(T,\rho)=\lim_{V\to\infty,\ N/V\to\rho}\frac{F(T,V,N)}{V}.$$

  When the limit exists, $f$ depends only on the temperature and density (or equivalent intensive data), yielding an equation of state through derivatives.

• Entropy density. With $S(U,V,N)$ the entropy , one similarly considers

  $$s(u,\rho)=\lim_{V\to\infty,\ U/V\to u,\ N/V\to\rho}\frac{S(U,V,N)}{V}.$$

• Pressure as a limit potential. In the grand canonical setting, the grand potential typically satisfies $\Omega(T,V,\mu)\approx -PV$ for large $V$, so the limiting pressure becomes a state function of $T$ and the chemical potential .

Physical interpretation. Thermodynamic-limit state functions are the “bulk” thermodynamic objects measured in macroscopic experiments: they are insensitive to microscopic boundary details and encode equilibrium properties of matter.

Structural properties.

• Homogeneity and Euler/Gibbs–Duhem structure. When extensivity holds, extensive potentials become (asymptotically) homogeneous of degree one in extensive variables, leading in the limit to identities such as the Euler relation and the Gibbs–Duhem relation .
• Convexity/concavity from stability. Under stability , limiting free-energy-like densities are convex in their natural variables, while entropy-like densities are concave; this mirrors energy convexity vs. entropy concavity .
• Ensemble consistency. In many settings, the existence and uniqueness of these limits underlies equivalence between canonical and grand canonical descriptions at the level of thermodynamic functions.