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| {{stub-general}}
| | The excess [[chemical potential]] is given by |
| The '''Widom test-particle method''', proposed by [[Benjamin Widom]] <ref>[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]</ref><ref>[http://dx.doi.org/10.1021/j100395a005 B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry '''86''' pp. 869-872 (1982)]</ref>, is an elegant, general [[Computer simulation techniques |simulation technique]] to obtain the excess [[chemical potential]] of a system. A so-called ''test particle'' is introduced in a [[Random numbers |random]] location, and <math>\Delta\Phi</math>, the difference in [[internal energy]] before and after the insertion, is computed. For [[Intermolecular pair potential |pairwise interactions]], this would become be the interaction potential energy between the randomly placed test particle and the <math>N</math> particles that the system is comprised of. The particle is not actually inserted, at variance with [[Monte Carlo in the grand-canonical ensemble|grand canonical Monte Carlo]]. | |
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| The excess chemical potential is given by
| | :<math>\mu^{ex} = -k_BT \ln \langle e^{-\Phi/k_bT}\rangle_N</math> |
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| :<math>\mu^{excess} = \mu -\mu^{ideal} = -k_BT \log \langle e^{-\Delta\Phi/k_BT}\rangle_N ,</math>
| | where <math>k_B</math> is the [[Boltzmann constant]], ''T'' is the [[temperature]], ''N'' is the number of particles within the system and <math>\Phi</math> is the [[intermolecular pair potential]]. |
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| where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]]. | |
| ==References== | | ==References== |
| <references/>
| | #[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)] |
| ;Related reading
| | #[http://dx.doi.org/10.1021/j100395a005 B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry '''86''' pp. 869 - 872 (1982)] |
| *[http://dx.doi.org/10.1080/002689798169104 David S. Corti "Alternative derivation of Widom's test particle insertion method using the small system grand canonical ensemble", Molecular Physics '''93''' pp. 417-420 (1998)]
| | #[http://dx.doi.org/10.1080/002689798169104 David S. Corti "Alternative derivation of Widom's test particle insertion method using the small system grand canonical ensemble", Molecular Physics '''93''' pp. 417-420 (1998)] |
| *[http://dx.doi.org/10.1063/1.4968039 David M. Heyes and Andrés Santos "Chemical potential of a test hard sphere of variable size in a hard-sphere fluid", Journal of Chemical Physics '''145''' 214504 (2016)]
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| [[category: computer simulation techniques]] | | [[category: computer simulation techniques]] |