Extensivity postulate

Definition (and physical meaning)

The extensivity postulate asserts that macroscopic thermodynamics can be formulated so that the fundamental state description becomes scale-invariant under replication of the system. Concretely, in the thermodynamic limit (where boundary effects are negligible compared with bulk), the thermodynamic entropy in the entropy representation can be taken to satisfy homogeneity of degree one in the extensive variables:

where $E$ is internal energy , $V$ is volume , and $N$ is particle number . Equivalently, other thermodynamic potentials (with the appropriate choice of independent variables) become homogeneous in their extensive arguments.

Physical interpretation: if you make $\lambda$ independent copies of the same macroscopic state and combine them without introducing significant interfacial effects, bulk properties “scale with size.” This is the content behind calling $E,V,N,S$ extensive variables .

Key consequences and relations

• Intensive variables are scale-invariant. If $S(E,V,N)$ is homogeneous of degree one, then derived intensive variables like temperature , pressure , and chemical potential are homogeneous of degree zero (they do not change under uniform scaling of $E,V,N$).

• Euler relation and Gibbs–Duhem. Homogeneity (a homogeneous function of degree one property) implies the Euler relation . In the entropy representation (with only $pV$ work),

  $$S=\frac{1}{T}E+\frac{p}{T}V-\frac{\mu}{T}N,$$

  and in the energy representation,

  $$E=TS-pV+\mu N.$$

  Differentiating Euler’s relation yields the Gibbs–Duhem relation , constraining how $T,p,\mu$ can vary.

• Connection to additivity. Extensivity is typically justified by (approximate) additivity together with negligible interfacial/boundary contributions in the thermodynamic limit .

When it can fail (usefully to remember)

Extensivity can break down when boundary or long-range interaction contributions remain comparable to bulk terms (e.g., strong surface effects, unscreened long-range forces, or constraints that prevent decomposition into weakly interacting parts). In such cases, thermodynamic potentials may acquire non-negligible subextensive terms, and the usual Euler/Gibbs–Duhem structure must be modified.