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Difference between revisions of "Stirling's approximation"

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:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
 
:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
 +
 +
after some further manipulation one arrives at
 +
 +
:<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>
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 +
where
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:<math>\frac{1}{12N+1} < \lambda_N < \frac{1}{12N}.</math>
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For example:
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{| border="1"
 +
|-
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| N || N! (exact) || N! (Stirling)  || Error (%)
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|-
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|5 ||  120  || 118.019168 || 1.016
 +
|-
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|6  || 720  ||  710.078185 || 1.014
 +
|-
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|7  || 5040  || 4980.39583  || 1.012
 +
|-
 +
|8  ||  40320 ||  39902.3955 || 1.010
 +
|-
 +
|9  ||  362880||  359536.873  || 1.009
 +
|-
 +
|10  || 3628800  ||  3598695.62  || 1.008
 +
|}
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 +
As one usually deals with number of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially  exact.
 
==Applications in statistical mechanics==
 
==Applications in statistical mechanics==
 
*[[Ideal gas Helmholtz energy function]]
 
*[[Ideal gas Helmholtz energy function]]
 
[[Category: Mathematics]]
 
[[Category: Mathematics]]

Revision as of 19:14, 4 November 2008

James Stirling (1692-1770, Scotland)

\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .

Because of Euler-MacLaurin formula

\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R ,

where B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.

Then, for large N,

\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .

after some further manipulation one arrives at

N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}

where

\frac{1}{12N+1} < \lambda_N < \frac{1}{12N}.

For example:

N N! (exact) N! (Stirling) Error (%)
5 120 118.019168 1.016
6 720 710.078185 1.014
7 5040 4980.39583 1.012
8 40320 39902.3955 1.010
9 362880 359536.873 1.009
10 3628800 3598695.62 1.008

As one usually deals with number of the order of the Avogadro constant (10^{23}) this formula is essentially exact.

Applications in statistical mechanics