Additivity postulate

Definition (and physical meaning)

The additivity postulate states that if a macroscopic system can be decomposed into subsystems $A$ and $B$ separated by an effective boundary so that interaction energies and correlations across the boundary are negligible at the macroscopic level, then extensive state variables add:

• $E=E_A+E_B$ for internal energy ,
• $V=V_A+V_B$ for volume ,
• $N=N_A+N_B$ for particle number , and, crucially for equilibrium reasoning, $S(E,V,N)=S_A(E_A,V_A,N_A)+S_B(E_B,V_B,N_B)$ (up to subextensive corrections due to the interface).

Physical interpretation: additivity is the macroscopic expression of “weak coupling at a distance.” Distant parts of a large system can be treated as contributing independently to bulk thermodynamic bookkeeping.

Additivity is closely tied to the existence of the thermodynamic limit and underlies the extensivity postulate for short-range interacting matter.

Equilibrium implications

Consider an isolated system composed of two additively coupled subsystems that can exchange energy. With $E=E_A+E_B$ fixed, additivity gives $S=S_A(E_A)+S_B(E_B)$, so maximizing total entropy yields

which is exactly the condition for thermal equilibrium (since $\partial S/\partial E = 1/T$ defines temperature ). Analogous entropy-maximization arguments produce the conditions for mechanical equilibrium and chemical equilibrium .

Connection to information-theoretic additivity

In statistical mechanics, it is often useful to compare thermodynamic entropy to dimensionless entropies such as Shannon entropy : both become additive when the underlying subsystems/variables are independent. The thermodynamic statement is physically grounded in negligible interfacial interactions and correlations, not merely a formal property of a formula.

Where additivity can break down

Additivity can fail when interactions across the boundary contribute at the same order as bulk terms (long-range forces, strong interfacial coupling, or persistent correlations). In such cases, treating the total entropy as a sum can miss macroscopic “interaction entropy” terms, and standard equilibrium derivations must be revisited.