Stirling’s log-factorial approximation

In my statistical physics course, I have frequently encountered the following approximation due to Stirling:

\ln(N!) \approx N \ln (N) -N

It’s very useful but my professor didn’t explain how good the approximation was. The derivation I found turns out to be very simple and so I can present it in a few lines here:

  1. Note that:\int \ln(x) dx = x \ln (x) -x \quad (1)
  2. Now, if we defineS = \sum_{n=1}^{N} \ln (n) \quad (2)
    we have an upper-Riemann sum with \Delta x =1 .
  3. So we basically have the following approximation:
    S = \sum_{n=1}^{N} \ln (n \Delta x)\Delta x \approx \int_{1}^{N} \ln(x) dx \quad (3)
  4. By the intermediate value theorem,\forall n \in\mathbb{N} \thinspace \exists c_n \in (n-1,n), S' =  \sum_{n=1}^{N} \ln (c_n \Delta x)\Delta x =\int_{1}^{N} \ln(x) dx \quad(4) where \Delta x =1 as defined previously.
  5. Let’s check how good this approximation is:|S - S'| = | \sum_{n=1}^{N} \ln (n) - \sum_{n=1}^{N} \ln (c_n) | \Rightarrow |S - S'| \leq | \sum_{n=1}^{N} \ln (n) -\ln (n+1)| = \ln (N+1) \quad (5)
  6. This error grows very slowly. In fact, if N = 10^{24} i.e. the number of molecules in a glass of water,| \ln(N!) - (N \ln (N) -N)| < 60 which is a minuscule error relative to the number of molecules.

Note: A pdf version of this blog post is available here.

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