Widom test-particle method
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Benjamin Widom proposed an elegant, general simulation technique to obtain the excess chemical potential of a system. A so-called test particle is introduced in a random location, and 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 \Delta\Phi} , the difference in internal energy before and after the insertion, is computed. For pairwise interactions, this would become be the interaction potential energy between the randomly placed test particle and the 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} particles that the system is comprised of. The particle is not actually inserted, at variance with grand canonical Monte Carlo.
The excess chemical potential is given by
- 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 \mu^{ex} = -k_BT \log \langle e^{-\Delta\Phi/k_bT}\rangle_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 k_B} is the Boltzmann constant and 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 T} is the temperature.
References
- B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics 39 pp. 2808-2812 (1963)
- B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry 86 pp. 869 - 872 (1982)
- 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)