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'''Stirling's approximation''' was invented by the Scottish mathematician James Stirling (1692-1770).
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'''Stirling's approximation''' is named after the Scottish mathematician James Stirling (1692-1770)<ref>J. Stirling "Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium", London (1730). English translation by J. Holliday "The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series" (1749)</ref>.
  
 
:<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>
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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  
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after some further manipulation one arrives at (apparently  Stirling's contribution was the prefactor of <math>\sqrt{2 \pi}</math>)
  
 
:<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>
 
:<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>
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|-  
 
|-  
 
| N || N! (exact) || N! (Stirling)  || Error (%)
 
| N || N! (exact) || N! (Stirling)  || Error (%)
 +
|-
 +
|1 ||  1  || 0.92213700 ||  8.44
 
|-  
 
|-  
 
|2 ||  2  || 1.91900435 ||  4.22
 
|2 ||  2  || 1.91900435 ||  4.22
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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  
 
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 approximately 0.083%.  
 
percentage error is approximately 0.083%.  
 +
==Gosper’s formula==
 +
Gosper’s formula <ref>[http://www.pnas.org/content/75/1/40 R. William Gosper, Jr. "Decision procedure for indefinite hypergeometric summation", PNAS '''75''' pp. 40-42 (1978)]</ref><ref>[http://dx.doi.org/10.1016/j.amc.2009.12.013  Cristinel Mortici "Best estimates of the generalized Stirling formula", Applied Mathematics and Computation '''215''' pp. 4044-4048 (2010)]</ref>:
 +
 +
:<math>n! \approx \sqrt{2 \pi \left( n + \frac{1}{6} \right)} \;  \left( \frac{n}{e} \right)^n</math>
 +
 +
Which results in:
 +
 +
{| border="1"
 +
|-
 +
| N || N! (exact) || N! (Gosper)
 +
|-
 +
|1 ||  1        || 0.99602180
 +
|-
 +
|2 ||  2        || 1.99736305
 +
|-
 +
|3 ||  6        || 5.99613535
 +
|-
 +
|4 ||  24        || 23.9908895
 +
|-
 +
|5 ||  120      || 119.970030
 +
|-
 +
|6  || 720      || 719.872829
 +
|-
 +
|7  || 5040      || 5039.33747
 +
|-
 +
|8  ||  40320    || 40315.9028
 +
|-
 +
|9  ||  362880  || 362850.646
 +
|-
 +
|10  || 3628800  || 3628560.82
 +
|}
 
==Applications in statistical mechanics==
 
==Applications in statistical mechanics==
 
*[[Ideal gas Helmholtz energy function]]
 
*[[Ideal gas Helmholtz energy function]]
 +
==References==
 +
<references/>
 
[[Category: Mathematics]]
 
[[Category: Mathematics]]

Latest revision as of 12:33, 31 January 2011

Stirling's approximation is named after the Scottish mathematician James Stirling (1692-1770)[1].

\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 (apparently Stirling's contribution was the prefactor of \sqrt{2 \pi})

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 (%)
1 1 0.92213700 8.44
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%.

Gosper’s formula[edit]

Gosper’s formula [2][3]:

n! \approx \sqrt{2 \pi \left( n + \frac{1}{6} \right)} \;  \left( \frac{n}{e} \right)^n

Which results in:

N N! (exact) N! (Gosper)
1 1 0.99602180
2 2 1.99736305
3 6 5.99613535
4 24 23.9908895
5 120 119.970030
6 720 719.872829
7 5040 5039.33747
8 40320 40315.9028
9 362880 362850.646
10 3628800 3628560.82

Applications in statistical mechanics[edit]

References[edit]

  1. J. Stirling "Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium", London (1730). English translation by J. Holliday "The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series" (1749)
  2. R. William Gosper, Jr. "Decision procedure for indefinite hypergeometric summation", PNAS 75 pp. 40-42 (1978)
  3. Cristinel Mortici "Best estimates of the generalized Stirling formula", Applied Mathematics and Computation 215 pp. 4044-4048 (2010)