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Poisson distribution from binomial using Stirling’s approximation

Posted by peeterjoot on January 29, 2013

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Question: Large N approximation of binomial distribution

In section 11.1 [1] it is stated that the binomial distribution

\begin{aligned}\rho(n) = \binom{N}{n} p^n (1 - p)^{N-n},\end{aligned} \hspace{\stretch{1}}(1.0.1)

has the large N approximation of a Poisson distribution

\begin{aligned}\rho(n) = \frac{(\alpha v)^n}{n!} e^{-\alpha v},\end{aligned} \hspace{\stretch{1}}(1.0.2)

where N/V = \alpha is the density and p = v/V, the probability (of finding the particle in a volume v of a total volume V in this case).

Show this.

Answer

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This is another Stirling’s approximation problem. With p = v \alpha/N, and working with log expansion of the N! and (N-n)! terms of the binomial coefficient we have

\begin{aligned}\ln \rho &\approx{\left({ N + \frac{1}{{2}} }\right)} \ln N - \not{{N}} - {\left({ N - n + \frac{1}{{2}} }\right)} \ln (N - n) + (\not{{N}} - n) - \ln n! + n \ln p + (N - n) \ln (1 - p) \\ &= {\left({ N + \frac{1}{{2}} }\right)} \ln N - {\left({ N - n + \frac{1}{{2}} }\right)} \ln (N - n) - n - \ln n! + n \ln \frac{v \alpha}{N} + (N - n) \ln {\left({1 - \frac{v \alpha}{N}}\right)} \\ &= {\left({ N + \frac{1}{{2}} - N + n - \frac{1}{{2}} -n }\right)} \ln N - {\left({ N - n + \frac{1}{{2}} }\right)} \ln {\left({1 - \frac{n}{N}}\right)} - n - \ln n! + n \ln v \alpha + (N - n) \ln {\left({1 - \frac{v \alpha}{N}}\right)} \\ &\approx\ln \frac{(v\alpha)^n}{n!} - n- {\left({ N - n + \frac{1}{{2}} }\right)} {\left({- \frac{n}{N} - \frac{1}{{2}}\frac{n^2}{N^2} }\right)} + (N - n) {\left({- \frac{v \alpha}{N} - \frac{1}{{2}} \frac{v^2 \alpha^2}{N^2}}\right)} \\ &= \ln \frac{(v\alpha)^n}{n!} - n+ n {\left({ 1 - \frac{n}{N} + \frac{1}{{2N}} }\right)} {\left({1 + \frac{1}{{2}}\frac{n}{N} }\right)} - v\alpha {\left({1 - \frac{n}{N} }\right)} {\left({1 + \frac{1}{{2}} \frac{v \alpha}{N}}\right)} \\ &\approx \ln \frac{(v\alpha)^n}{n!} - n+ n - v\alpha\end{aligned} \hspace{\stretch{1}}(1.0.3)

Here all the 1/N terms have been dropped, and we are left with

\begin{aligned}\ln \rho \approx \ln \frac{(v\alpha)^n}{n!} - v\alpha,\end{aligned} \hspace{\stretch{1}}(1.0.4)

which is the logarithm of the Poisson distribution as desired.

References

[1] S.K. Ma. Statistical Mechanics. World Scientific, 1985. ISBN 9789971966072. URL http://books.google.ca/books?id=3XyA0SYK4XIC.

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