Stirling's approximation: Difference between revisions

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]

Ideal gas Helmholtz energy function

References[edit]

↑ 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)
↑ R. William Gosper, Jr. "Decision procedure for indefinite hypergeometric summation", PNAS 75 pp. 40-42 (1978)
↑ Cristinel Mortici "Best estimates of the generalized Stirling formula", Applied Mathematics and Computation 215 pp. 4044-4048 (2010)

[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)

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-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''.
-:<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]]

Stirling's approximation: Difference between revisions

Latest revision as of 12:33, 31 January 2011

Gosper’s formula[edit]

Applications in statistical mechanics[edit]

References[edit]

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