Stirling's Formula

Asymptotic approximation for n! and log n!, used for entropy and large-N counting in statistical mechanics.

Stirling’s Formula

Stirling’s formula is a basic asymptotic tool for combinatorial and phase-space counting, and it is frequently used when connecting microscopic counting to Boltzmann entropy and to information-theoretic quantities like Shannon entropy .

Statement

As $n\to\infty$ ,

n! \sim \sqrt{2\pi n}\,\Big(\frac{n}{e}\Big)^n.

Equivalently,

\log(n!) = n\log n - n + \tfrac12\log(2\pi n) + o(1).

A common quantitative refinement is: for every integer $n\ge 1$ there exists $\theta_n\in(0,1)$ such that

n! = \sqrt{2\pi n}\,\Big(\frac{n}{e}\Big)^n \exp\!\Big(\frac{\theta_n}{12n}\Big).

In particular,

\log(n!) = n\log n - n + \tfrac12\log(2\pi n) + O\!\Big(\frac{1}{n}\Big).

Key hypotheses and conclusions

Hypotheses

$n\in\mathbb{N}$ and $n\to\infty$ .

Conclusions

Accurate leading-order and next-order asymptotics for $n!$ and $\log(n!)$ .
Enables asymptotics for multinomial coefficients; e.g. leading terms produce entropy-like functionals.

Proof idea / significance

One proof route uses integral bounds for $\log(n!)=\sum_{k=1}^n \log k$ compared to $\int_1^n \log x\,dx$ , plus a refinement (e.g. Euler–Maclaurin) to obtain the $\tfrac12\log n$ correction and the constant $\sqrt{2\pi}$ . Another route uses $\Gamma(n+1)=\int_0^\infty t^n e^{-t}\,dt$ and applies a Laplace/saddle-point argument (see saddle-point asymptotics ).

In statistical mechanics, Stirling’s formula converts factorial growth in state counting (e.g. indistinguishable particles, multinomial occupations) into extensive terms of order $n$ plus subextensive corrections. This is one mechanism by which entropy densities emerge from microscopic counting.