Annus MirabilisEinstein in 1905
Reading preferences

Reading preferences

Keep every explanation and worked example on the page. Nothing moves or plays by itself, and the larger instruments load only when you ask.

Line length
Type size
Contrast
Paragraph spacing

light quanta

The equation explained

φ1(W1)=Clg⁡(W1)+konst.,φ2(W2)=Clg⁡(W2)+konst.,φ(W)=Clg⁡(W)+konst.\begin{aligned} \varphi_1(W_1) &= C \lg(W_1) + \text{konst.}, \\ \varphi_2(W_2) &= C \lg(W_2) + \text{konst.}, \\ \varphi(W) &= C \lg(W) + \text{konst.} \end{aligned}
In wordsEach entropy function is the same constant C times the natural logarithm of the probability, plus a constant: for the first system, for the second, and for the two together.OverviewThe only functions with that property are a constant times the logarithm, plus a constant, and the multiplying constant C is the same for both systems and for the pair. So C is a universal constant.Full explanationThe paper goes from the preceding line to this one with the words "and from this, finally", without showing the step. The steps below fill it in for functions that can be differentiated, which the paper takes for granted. The multiplying constant CC turns out to be the same in all three lines. Since the two systems were any two systems, the paper concludes that CC is a universal constant. The added constants may differ, and the one for the pair is the sum of the other two. The paper writes lg⁡\lg for the natural logarithm.Every stepW2dφdW=dφ1dW1W_2 \frac{d \varphi}{d W} = \frac{d \varphi_1}{d W_1}Differentiate the preceding display with respect to W1W_1, holding W2W_2 fixed. By the chain rule the left side gives the rate of change of φ\varphi, taken where W=W1⋅W2W = W_1 \cdot W_2, times W2W_2. The paper does not show this or the following steps.W1dφdW=dφ2dW2W_1 \frac{d \varphi}{d W} = \frac{d \varphi_2}{d W_2}The same with respect to W2W_2, holding W1W_1 fixed.WdφdW=W1dφ1dW1=W2dφ2dW2W \frac{d \varphi}{d W} = W_1 \frac{d \varphi_1}{d W_1} = W_2 \frac{d \varphi_2}{d W_2}Multiply the first line by W1W_1 and the second by W2W_2, and use W1⋅W2=WW_1 \cdot W_2 = W.W1dφ1dW1=CW_1 \frac{d \varphi_1}{d W_1} = CThe middle term depends only on W1W_1 and the last only on W2W_2. The two probabilities can be changed independently, so each term must equal one and the same constant, which is called CC.dφ1dW1=CW1\frac{d \varphi_1}{d W_1} = \frac{C}{W_1}Divide by W1W_1.φ1(W1)=Clg⁡(W1)+konst.\varphi_1(W_1) = C \lg(W_1) + \text{konst.}The function whose rate of change is CC divided by the variable is CC times the natural logarithm, plus a constant. The same reasoning gives φ2\varphi_2 and φ\varphi, with the same CC: the printed lines.Also

Read it in the paper