A Statistical View of Entropy — Question 4

Question 4

Relative occupancy fluctuations shrink as the particle number grows

For $N$ independent particles in equal box halves, write $n=\sum_{i=1}^N X_i$, where $X_i=1$ if particle $i$ is on the left and 0 otherwise. Each outcome has probability $1/2$. Use independence to add variances.

1. Derive the mean and standard deviation of $n$. Evaluate both for $N=10^8$, including the standard deviation of the fraction $n/N$.

2. Find the minimum integer $N$ for the standard deviation divided by mean left occupancy to be no greater than 1 percent. Distinguish this criterion from a guarantee on every observation.

Question 4 - Solution

Relative occupancy fluctuations shrink as the particle number grows

(a) Method and calculation

$$\langle X_i\rangle=1/2,\quad\operatorname{Var}(X_i)=1/2-(1/2)^2=1/4.$$$$\langle n\rangle=N/2,\quad\sigma_n=\sqrt N/2;\quad N=10^8:\ \langle n\rangle=5\times10^7,\ \sigma_n=5000.$$$$\sigma_{n/N}=1/(2\sqrt N)=5\times10^{-5}.$$

(b) Interpretation and further calculation

$$\frac{\sigma_n}{\langle n\rangle}=\frac1{\sqrt N}\le.010\quad\Rightarrow\quad N\ge10^4.$$$$\sigma_n\propto\sqrt N,\qquad\sigma_n/\langle n\rangle\propto N^{-1/2}.$$Absolute fluctuations grow while relative fluctuations shrink. A one-standard-deviation criterion is not a bound on all possible outcomes.

Physical check

The diagram shows the decreasing scaling, not a normal approximation or a statement that fluctuations vanish exactly.