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# Difference between revisions of "Stirling's approximation"

Carl McBride (talk | contribs) (Added a table) |
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| N || N! (exact) || N! (Stirling) || Error (%) | | N || N! (exact) || N! (Stirling) || Error (%) | ||

|- | |- | ||

− | | | + | |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 |

|- | |- | ||

− | |10 || 3628800 || 3598695.62 || | + | |8 || 40320 || 39902.3955 || 1.05 |

+ | |- | ||

+ | |9 || 362880|| 359536.873 || 0.93 | ||

+ | |- | ||

+ | |10 || 3628800 || 3598695.62 || 0.84 | ||

|} | |} | ||

− | As one usually deals with number of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially exact. | + | As one usually deals with number 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, where the | ||

+ | percentage error is less than . | ||

==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 14:08, 5 November 2008

James Stirling (1692-1770, Scotland)

Because of Euler-MacLaurin formula

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 (%) |

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 |

As one usually deals with number 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, where the percentage error is less than .