Difference between revisions of "Widom test-particle method"

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{{stub-general}}
 
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The excess [[chemical potential]] is given by
+
[[Benjamin Widom]] proposed an elegant, general method to obtain
 +
the excess [[chemical potential]] of a system that is being
 +
simulated. A ''test particle'' is introduced in a random
 +
location, and <math>\Delta\Phi</math>, the difference
 +
in [[internal energy]] before and after the insertion,
 +
is computed. (For pairwise interactions, this would
 +
just be the interaction potential energy between the randomly
 +
placed test particle and the ''N'' 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 MC ]]).
  
:<math>\mu^{ex} = -k_BT \ln \langle e^{-\Phi/k_bT}\rangle_N</math>
+
The excess chemical potential is given by
  
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 interaction potential energy between the randomly placed test particle and the ''N'' particles that the system is comprised of.
+
:<math>\mu^{ex} = -k_BT \log \langle e^{-\Delta\Phi/k_bT}\rangle_N ,</math>
 +
 
 +
where <math>k_B</math> is the [[Boltzmann constant]] and ''T'' is the [[temperature]].
 
==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)]
 
#[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)]

Revision as of 22:04, 29 January 2008

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Benjamin Widom proposed an elegant, general method to obtain

the excess chemical potential of a system that is being simulated. A test particle is introduced in a random location, and \Delta\Phi, the difference in internal energy before and after the insertion, is computed. (For pairwise interactions, this would just be the interaction potential energy between the randomly placed test particle and the N particles that the system is comprised of.) The particle is not actually inserted (at variance with grand canonical MC ).

The excess chemical potential is given by

\mu^{ex} = -k_BT \log \langle e^{-\Delta\Phi/k_bT}\rangle_N ,

where k_B is the Boltzmann constant and T is the temperature.

References

  1. B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics 39 pp. 2808-2812 (1963)
  2. B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry 86 pp. 869 - 872 (1982)
  3. 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)