Stirling's approximation: Difference between revisions

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James Stirling (1692-1770, Scotland)
'''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>


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>


where ''B''<sub>1</sub> = &minus;1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = &minus;1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = &minus;1/30, ... are the [http://en.wikipedia.org/wiki/Bernoulli_numbers Bernoulli numbers], and ''R'' is an error term which is normally small for suitable values of ''p''.
where ''B''<sub>1</sub> = &minus;1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = &minus;1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = &minus;1/30, ... are the [http://en.wikipedia.org/wiki/Bernoulli_numbers Bernoulli numbers], and ''R'' is an error term which is normally small for suitable values of ''p''.


:<math>~\approx \int_1^N \ln x dx</math>
Then, for large ''N'',


:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>


:<math>~= \left[ x \ln x - x \right]_1^N</math>
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 \ln N -N +1</math>
where


Thus, for large ''N''
:<math>\frac{1}{12N+1} < \lambda_N < \frac{1}{12N}.</math>


:<math>\ln N! \approx  N \ln N -N</math>
For example:
 
{| border="1"
|-
| 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 ]](<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;  for N=100 the
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==
*[[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].

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 (apparently Stirling's contribution was the prefactor of )

where

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 () 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]:

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)