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# Difference between revisions of "Stirling's approximation"

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− | James Stirling (1692-1770 | + | '''Stirling's approximation''' was invented by the Scottish mathematician James Stirling (1692-1770). |

:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math> | :<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math> | ||

− | + | using [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula] one has | |

:<math>\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 ,</math> | :<math>\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 ,</math> | ||

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|- | |- | ||

| N || N! (exact) || N! (Stirling) || Error (%) | | N || N! (exact) || N! (Stirling) || Error (%) | ||

+ | |- | ||

+ | |2 || 2 || 1.91900435 || 4.22 | ||

|- | |- | ||

|3 || 6 || 5.83620959 || 2.81 | |3 || 6 || 5.83620959 || 2.81 | ||

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|} | |} | ||

− | + | When one is dealing with numbers of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially exact. | |

− | In [[Computer simulation techniques | computer simulations]] the number of atoms or molecules (N) is invariably greater than 100 | + | In [[Computer simulation techniques | computer simulations]] the number of atoms or molecules (N) is invariably greater than 100; for N=100 the |

− | percentage error is | + | percentage error is approximately 0.083%. |

==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 15:16, 5 November 2008

**Stirling's approximation** was invented by the Scottish mathematician James Stirling (1692-1770).

using Euler-MacLaurin formula one has

where *B*_{1} = −1/2, *B*_{2} = 1/6, *B*_{3} = 0, *B*_{4} = −1/30, *B*_{5} = 0, *B*_{6} = 1/42, *B*_{7} = 0, *B*_{8} = −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*,

after some further manipulation one arrives at

where

For example:

N | N! (exact) | N! (Stirling) | Error (%) |

2 | 2 | 1.91900435 | 4.22 |

3 | 6 | 5.83620959 | 2.81 |

4 | 24 | 23.5061751 | 2.10 |

5 | 120 | 118.019168 | 1.67 |

6 | 720 | 710.078185 | 1.40 |

7 | 5040 | 4980.39583 | 1.20 |

8 | 40320 | 39902.3955 | 1.05 |

9 | 362880 | 359536.873 | 0.93 |

10 | 3628800 | 3598695.62 | 0.84 |

When one is dealing with numbers of the order of the Avogadro constant () this formula is essentially exact. In computer simulations the number of atoms or molecules (N) is invariably greater than 100; for N=100 the percentage error is approximately 0.083%.