= Additivity of Boltzmann entropy
{title2=$\Omega_{AB}=\Omega_A\Omega_B,\quad S_{AB}=S_A+S_B$}
For two subsystems with independently specified <macrostates>, suppose every allowed <microstate> of $A$ can be paired with every allowed <microstate> of $B$. Their allowed set is a <Cartesian product>. With equal <probability> for each pair, <Boltzmann's entropy formula> gives
$$
\Omega_{AB}=\Omega_A\Omega_B,\qquad
S_{AB}=k_B\ln(\Omega_A\Omega_B)=S_A+S_B.
$$
For example, take two <independent> copies of the toy model, each with two labels choosing between its two boxes. Each copy has four configurations and <Boltzmann entropy> $k_B\ln4$. Combining their <independent> labels gives sixteen configurations and <Boltzmann entropy> $k_B\ln16=2k_B\ln4$. This explains why <entropy> adds even though microscopic counts multiply. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node57.html[MIT, §7.4, equation 7.17: independent state counts multiply]{ref}
The product assumption matters. A constraint shared between the subsystems can forbid some pairs; for example, fixing only their combined <energy> generally requires summing over its possible divisions. See <entropy maximization under thermal contact> for how <thermal equilibrium> selects the dominant division. Statistical <independence> cannot be inferred merely from drawing a boundary between two parts.
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