= Extensive quantity
= Extensive variable
{synonym}
Two related properties of an <extensive quantity> must be distinguished. Additivity says that a quantity $X$ for independent subsystems A and B obeys
$$
X_{A\cup B}=X_A+X_B.
$$
The subsystems may be in different states. Extensivity expressed through a <homogeneous function> says that the equilibrium fundamental relation obeys
$$
U(\lambda S,\lambda V,\lambda n)=\lambda U(S,V,n),\qquad \lambda>0,
$$
with composition held fixed. This is degree-one <homogeneity> along a replication ray; it does not say that <internal energy> is a <linear function> of arbitrary independent changes of <entropy>, <volume (thermodynamics)> and <amount of substance>.
Additivity gives integer replication by repeated addition. Subdividing a uniform bulk sample into $b$ equivalent parts gives rational scaling $a/b$; continuity then extends it to positive real factors. This argument requires negligible boundary contributions and a consistently prepared state. Conversely, a degree-one <homogeneous function> need not be additive between different rays: $f(x,y)=\sqrt{x^2+y^2}$ has $f(\lambda x,\lambda y)=\lambda f(x,y)$, but $f(1,0)+f(0,1)=2\ne\sqrt2=f(1,1)$. Thus a scaling identity alone cannot justify summing arbitrary states.
For finite samples, the <surface correction to thermodynamic free energy> can matter. Under geometric enlargement with <volume> multiplied by $\lambda$, <surface area> grows as $\lambda^{2/3}$, so a <surface energy> term is not a bulk <extensive quantity>. Replicating separated droplets retains every original surface, whereas merging them changes the total <surface area>. Likewise, appreciable mutual <gravitational potential energy> makes interacting subsystems fail the independent-copy premise. The <thermodynamic limit> explains when bulk scaling becomes accurate; it does not remove these qualifications for every small or strongly interacting system.
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