Editing Stirling's approximation
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'''Stirling's approximation''' | '''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> | ||
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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 | after some further manipulation one arrives at | ||
:<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 (%) | ||
|- | |- | ||
|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%. | ||
==Applications in statistical mechanics== | ==Applications in statistical mechanics== | ||
*[[Ideal gas Helmholtz energy function]] | *[[Ideal gas Helmholtz energy function]] | ||
[[Category: Mathematics]] | [[Category: Mathematics]] |