The Man in a Tailcoat

Abstract

Yesterday around midnight I had an interesting conversation with an elegant man. With a top hat and a curious glass walking cane, he was standing on the parapet of a bridge, looking down at the dark river below his feet.

Appendix—H Versus S via Log-Factorial Approximation

In general, we can rewrite the 1-block entropy $H^{(n_0 + n_1)}$ of Eq. (16.8) (where $n_0+n_1 = 600$) as:

$$\begin{aligned} H^{n_0 + n_1}((n_0, n_1))= & {} (n_0 + n_1)\Bigg (-\frac{n_0}{n_0 + n_1} log_2 \frac{n_0}{n_0 + n_1} -\frac{n_1}{n_0 + n_1} log_2 \frac{n_1}{n_0 + n_1}\Bigg ) \nonumber \\= & {} (n_0 + n_1) log_2(n_0 + n_1) - n_0 log_2(n_0) - n_1 log_2(n_1) \end{aligned}$$

(16.11)

Using base-2 logarithms and proportionality factor 1, the Boltzmann entropy of Eq. (16.10), relative to tuples with bit count $(n_0, n_1)$, becomes:

$$\begin{aligned} S((n_0, n_1))= & {} log_2 \frac{(n_0+n_1)!}{n_0! n_1!} \nonumber \\= & {} log_2e * (log((n_0+n_1)!) - log(n_0!) - log(n_1!)) \end{aligned}$$

(16.12)

where log() denotes the base e natural logarithm. By using the following approximation of the log-factorial function [3]:

$$\begin{aligned} log(n!) \approx \Bigg (n + \frac{1}{2}\Bigg ) log(n+1) - (n+1) + \frac{1}{2} log(2\pi ) + \frac{1}{12(n+1)} \end{aligned}$$

(16.13)

we further develop Eq. (16.12) as follows:

$$\begin{aligned} S((n_0, n_1))\approx & {} \Bigg (n_0 + n_1 + \frac{1}{2}\Bigg ) log_2(n_0 + n_1 + 1) - \Bigg (n_0 + \frac{1}{2}\Bigg ) log_2(n_0 + 1) \nonumber \\&- \Bigg (n_1 + \frac{1}{2}\Bigg ) log_2(n_1 + 1) + SS(n_0, n_1) \end{aligned}$$

(16.14)

where:

$$\begin{aligned} SS(n_0, n_1)= & {} log_2(e) - \frac{1}{2} log_2(2\pi ) + \frac{log_2(e)}{12}\Bigg (\frac{1}{n_0 + n_1 + 1} - \frac{1}{n_0 + 1} - \frac{1}{n_1 + 1}\Bigg ) \nonumber \\= & {} 0.117 + 0.120\Bigg (\frac{1}{n_0 + n_1 + 1} - \frac{1}{n_0 + 1} - \frac{1}{n_1 + 1}\Bigg ) \end{aligned}$$

(16.15)

Clearly the contribution of $SS(n_0, n_1)$ is negligible, while the first three terms of the approximated $S((n_0, n_1))$ of Eq. (16.14) parallel closely the three terms in the expansion of $H^{n_0+n_1}$ provided by Eq. (16.11). This explains the closeness of the two plots in Fig. 16.5-left.