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Difference between revisions of "Stirling's approximation"

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James Stirling (1692-1770, Scotland)
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'''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>
  
Because of [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula]
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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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|}
 
|}
  
As one usually deals with number of the order of the [[Avogadro constant ]](<math>10^{23}</math>)  this formula is essentially exact.
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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, where the  
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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 less than .  
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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).

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

using Euler-MacLaurin formula one has

\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 (%)
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 (10^{23}) 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%.

Applications in statistical mechanics