Stirling's approximation: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
mNo edit summary
m (Slight tidy.)
Line 1: Line 1:
James Stirling (1692-1770, Scotland)
'''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]
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>
Line 26: Line 26:
|-  
|-  
| 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
Line 44: Line 46:
|}
|}


As one usually deals with number of the order of the [[Avogadro constant ]](<math>10^{23}</math>)  this formula is essentially exact.
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  
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 .  
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 14:16, 5 November 2008

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

using Euler-MacLaurin formula one has

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,

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%.

Applications in statistical mechanics