Superadditivity of Entropy

This lemma is most naturally formulated for the microcanonical (Boltzmann) entropy , and it encodes the thermodynamic intuition behind thermodynamic entropy being extensive.

Statement (microcanonical form)

Consider two subsystems with disjoint degrees of freedom (e.g. two disjoint regions $\Lambda_1,\Lambda_2$) whose Hamiltonian is additive when they are decoupled:

Let $\Omega_\Lambda(E,\Delta)$ denote the number (or volume, in phase space) of microstates in an energy window $[E,E+\Delta]$, and define the windowed microcanonical entropy

Then for any $E_1,E_2$ and any $\Delta>0$,

and therefore

Key hypotheses and conclusions

Hypotheses

• The two subsystems are independent/decoupled so that the composite Hamiltonian is additive.
• Entropy is defined by logarithmic counting/volume of microstates (Boltzmann form).

Conclusions

• Superadditivity: the entropy of the union is at least the sum of entropies of parts at matching conserved totals.
• By optimizing over the split $E=E_1+E_2$, one obtains the familiar inequality $S_{\Lambda_1\cup\Lambda_2}(E,\Delta)\ge \sup_{E_1+E_2=E}\big(S_{\Lambda_1}(E_1,\Delta)+S_{\Lambda_2}(E_2,\Delta)\big)$, which is a precursor to concavity of the entropy density.

Proof idea / significance

If $x_1$ is a microstate of subsystem 1 with energy in $[E_1,E_1+\Delta]$ and $x_2$ is a microstate of subsystem 2 with energy in $[E_2,E_2+\Delta]$, then the product state $(x_1,x_2)$ is a microstate of the composite system with energy in $[E_1+E_2,E_1+E_2+\Delta]$. This injective map from pairs of microstates to composite microstates gives the multiplicative lower bound on $\Omega$, and taking $\log$ yields superadditivity.

For weakly interacting short-range systems, a variant holds up to boundary corrections (analogous to subadditivity for the partition function ): interactions across an interface contribute only $O(|\partial|)$ to the energy accounting, which is negligible per unit volume in the thermodynamic limit.

Superadditivity (or approximate superadditivity) is a standard route to existence of an entropy density via the superadditive form of Fekete's lemma , and it underlies the concavity/stability properties expected of equilibrium entropy.