Gauss and Poisson. (Basic)
In section 3.2.1, we calculated the probability distribution for having $n=N_{0}+m$ particles on the right-hand half of a box of volume $2 \mathrm{~V}$ with $2 \mathrm{~N}_{0}$ total particles. In section $10.3$ we will want to know the number fluctuations of a small subvolume in an infinite system. Studying this also introduces the Poisson distribution.
Let's calculate the probability of having $n$ particles in a subvolume $V$, for a box with total volume $K V$ and a total number of particles $T=K N_{0 .}$ For $K=2$ we will derive our previous result, equation 3.13, including the prefactor. As $K \rightarrow \infty$ we will derive the infinite volume result.
(a) Find the exact formula for this probability: $n$ particles in $V$, with total of $T$ particles in $K V .$ (Hint: What is the probability that the first $n$ particles fall in the subvolume $V$, and the remainder $T-n$ fall outside the subvolume $(K-1) V ?$ How many ways are there to pick $n$ particles from $T$ total particles?)
The Poisson probability distribution
$$
\rho_{n}=a^{n} e^{-a} / n !
$$
arises in many applications. It arises whenever there is a large number of possible events $T$ each with a small probability $a / T ;$ the number of cars passing a given point during an hour on a mostly empty street, the number of cosmic rays hitting in a given second, etc.
(b) Show that the Poisson distribution is normalized: $\sum_{n} \rho_{n}=1 .$ Calculate the mean of the distribution $\langle n\rangle$ in terms of a. Calculate the variance (standand deviation squared) $\left\langle(n-\langle n))^{2}\right\rangle$
(c) As $K \rightarrow \infty$, show that the probability that $n$ particles fall in the subvolume $V$ has the Poisson distribution 3.66. What is a? (Hint: You'll need to use the fact that $e^{-a}=\left(e^{-1 / K}\right)^{R \alpha} \rightarrow(1-1 / K)^{K a}$ as $K \rightarrow \infty$, and the fact that $n \ll T_{-}$Here don't assume that $n$ is large: the Poisson distribution is valid even if there are only a few events.).
From parts (b) and (c), you should be able to conclude that the variance in the number of particles found in a volume $V$ inside an infinite system should be equal to $N_{0}$. the expected number of particles in the volume:
$$
\left\langle(n-(n))^{2}\right\rangle=N_{0}
$$
This is twice the squared fluctuations we found for the case where the volume $V$ was half of the total volume, equation 3.13. That makes sense, since the particles can fluctuate more freely in an infinite volume than in a doubled volume.
If $N_{0}$ is large, the probability $P_{m}$ that $N_{0}+m$ particles lie inside our volume will be Gaussian for any $K$. (As a special case, if $a$ is large the Poisson distribution is well approximated as a Gaussian.) Let's derive this distribution for all $K$. First, as in section $3.2 .1$, let's use the weak
form of Stirling's approximation, equation $3.10$ dropping the square root: $n ! \sim(\bar{n} / e)^{n}$.
(d) Using your result from part (a), write the exact formula for $\log \left(P_{m}\right)$. Apply the weak form of Stirling's formula. Expand your result around $m=0$ to second order in $m$, and show that $\log \left(P_{m}\right) \approx-m^{2} / 2 \sigma_{K}^{2}$, giving a Gaussian form
$$
P_{m} \sim e^{-m^{2} / 2 \sigma_{K}^{2}}
$$
What is $\sigma_{K}$ ? In particular, what is $\sigma_{2}$ and $\sigma_{\infty}$ ? Your result for $\sigma_{2}$ should agree with the calculation in section 3.2.1, and your result for $\sigma_{\infty}$ should agree with equation $3.67$.
Finally, we should address the normalization of the Gaussian. Notice that the ratio of the strong and weak forms of Stirling's formula, (equation 3.10) is $\sqrt{2 \pi n}$. We need to use this to produce the normalization $\frac{1}{\sqrt{2 \pi \sigma} K}$ of our Gaussian.
(e) In terms of $T$ and $n$, what factor would the squareroot term have contributed if you had kept it in Stirling's formula going from part (a) to part (d)? (It should look like a ratio involving three terms like $\sqrt{2 \pi X}$.) Show from equation $3.68$ that the fluctuations are small, $m=$ $n-N_{0} \ll N_{0}$ for large $N_{0}$. Ignoring these fluctuations, set $n=N_{0}$ in your factor, and give the prefactor multiplying the Gaussian in equation 3.68. (Hint: your answer should be normalized.)