Annus MirabilisEinstein in 1905
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The equation explained

φ(W1⋅W2)=φ1(W1)+φ2(W2)\varphi(W_1 \cdot W_2) = \varphi_1(W_1) + \varphi_2(W_2)
In wordsThe entropy function, applied to the product of the two probabilities, equals the first system's entropy function of its probability plus the second system's entropy function of its probability.OverviewPut the three preceding lines together: the entropy function of a product of probabilities equals the sum of the two entropy functions of the factors.Full explanationThis is the preceding three lines combined. The entropy of the pair is φ\varphi of its probability, that probability is W1⋅W2W_1 \cdot W_2, and the entropy is also φ1(W1)+φ2(W2)\varphi_1(W_1) + \varphi_2(W_2). The result is an equation for unknown functions: whatever φ\varphi is, applied to a product it must give a sum. The logarithm has that property, since the logarithm of a product is the sum of the logarithms. The next display states that the logarithm, times a constant and plus a constant, is what the functions must be.Every stepφ(W)=S1+S2\varphi(W) = S_1 + S_2The entropy of the pair, written both ways (the second display).S1+S2=φ1(W1)+φ2(W2)S_1 + S_2 = \varphi_1(W_1) + \varphi_2(W_2)Each system's entropy as a function of its own probability (the first display).φ(W1⋅W2)=φ1(W1)+φ2(W2)\varphi(W_1 \cdot W_2) = \varphi_1(W_1) + \varphi_2(W_2)Put in W=W1⋅W2W = W_1 \cdot W_2 from the third display: the printed line.Also

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