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Stirling’s like approximation for log factorial

Posted by peeterjoot on December 24, 2012

Motivation

Back in my undergrad days, I rarely took notes, since most of what we covered was in the text books. The only notes I’d taken from the course where we did some thermodynamics was half an $8\, 1/2 \times 11$ scrap paper with a simple Stirling like approximation for $\ln N!$ (sitting in my old text [1]). Here it is

Guts

First write the factorial in product form

\begin{aligned}N! = \Pi_{k = 0}^{N-1} (N - k).\end{aligned} \hspace{\stretch{1}}(1.2.1)

so that it’s logarithm is a sum

\begin{aligned}\ln N! = \sum_{k = 0}^{N-1} \ln( N - k ).\end{aligned} \hspace{\stretch{1}}(1.2.2)

We can now derivatives of both sides with respect to $N$, ignoring the fact that $N$ is a discrete variable. For the left hand side, writing $f = N!$, we have

\begin{aligned}\frac{d \ln f }{dN} = \frac{1}{{f}} \frac{d f}{dN}.\end{aligned} \hspace{\stretch{1}}(1.2.3)

Now for the right hand side

\begin{aligned}\frac{d}{dN} \sum_{k = 0}^{N-1} \ln( N - k ) = \sum_{k = 0}^{N-1} \frac{1}{{ N - k }}\approx\int_{0}^{N-1} \frac{dk}{ N - k }= \int_{N}^{1} \frac{-du}{ u }= \ln N - \ln 1 = \ln N\end{aligned} \hspace{\stretch{1}}(1.2.4)

Merging 1.2.3 and 1.2.4 into two differentials, and integrating we have

\begin{aligned}\frac{df}{f} \approx \ln N dN.\end{aligned} \hspace{\stretch{1}}(1.2.5)

Observing that

\begin{aligned}(x \ln x -x)' = x \frac{1}{{x}} + \ln x - 1 = \ln x,\end{aligned} \hspace{\stretch{1}}(1.2.6)

we have

\begin{aligned}\ln N! \approx N \ln N - N + C.\end{aligned} \hspace{\stretch{1}}(1.2.7)

Even though we are interested in large $N$, we note that with $N = 1$ we have

\begin{aligned}0 = -1 + C,\end{aligned} \hspace{\stretch{1}}(1.2.8)

but this is small compared to $N$ for large $N$, so we have

\begin{aligned}\boxed{\ln N! \approx \ln N^N - N}\end{aligned} \hspace{\stretch{1}}(1.2.9)

Expontiating, this is

\begin{aligned}\boxed{N! \approx N^N e^{-N}}.\end{aligned} \hspace{\stretch{1}}(1.2.10)

I think that our Professor’s point at the time was that when we only care about the logarithm of $N!$ we can get away with 1.2.9, and can avoid the complexity and care required to do a proper Stirling’s approximation.

References

[1] C. Kittel and H. Kroemer. Thermal physics. WH Freeman, 1980.