Homogeneous function of degree one

A function $f(x_1,\dots,x_n)$ is homogeneous of degree one if, for all $\lambda>0$,

In thermodynamics, degree-one homogeneity is the mathematical expression of extensivity: doubling the size of a system (in a sense that scales all extensive variables together) doubles extensive thermodynamic potentials.

Physical interpretation (thermodynamics). For a simple macroscopic system obeying the extensivity postulate and additivity postulate , state functions like the entropy in the entropy fundamental relation or the internal energy in the energy fundamental relation are modeled as degree-one homogeneous in their extensive arguments: $S(U,V,N,\dots)$ and $U(S,V,N,\dots)$. This approximation is most accurate in the thermodynamic limit and can fail when surface terms, long-range interactions, or finite-size effects are significant.

Key consequences.

• Euler relation. If $U(S,V,N,\dots)$ is differentiable and degree-one homogeneous, then Euler’s theorem gives the Euler relation $U = TS - PV + \mu N + \cdots$, where $T,P,\mu$ are the conjugate intensive variables defined by derivatives of $U$.
• Gibbs–Duhem relation. Differentiating the Euler relation and comparing with $dU = T\,dS - P\,dV + \mu\,dN + \cdots$ yields the Gibbs–Duhem relation , constraining how intensive variables can change together.
• Intensive variables are scale-invariant. Under the same scaling of extensive variables, conjugate intensive variables such as temperature , pressure , and chemical potential remain unchanged (consistent with the definition of an intensive variable ).