Annus MirabilisEinstein in 1905
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The equation explained

S=S1+S2=φ(W)S = S_1 + S_2 = \varphi(W)
In wordsThe entropy of the two systems taken as one is the sum of their two entropies, and it is also some function of the probability of the combined system's state.OverviewTaken together as one system, the pair has an entropy that is the sum of the two entropies, and by the same principle it is also a function of the pair's probability.Full explanationRegard the two systems as a single system with entropy SS and probability WW. Entropy adds for systems that do not act on each other, so SS is the sum S1+S2S_1 + S_2. And the pair is a system in its own right, so the principle of the preceding line applies to it too: its entropy is some function φ\varphi of its probability. The next line says how WW is built from W1W_1 and W2W_2, and together the three lines fix what the function must be.Every stepS=S1+S2S = S_1 + S_2The two systems do not interact, so the entropy of the pair is the sum of their entropies.S=φ(W)S = \varphi(W)The pair is itself a system, so its entropy is a function of the probability of its state. The function φ\varphi is not assumed to be either system's own function.Also

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