Widom test-particle method

The excess chemical potential is given by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mu ^{ex}=-k_{B}T\ln \langle e^{-\Phi /k_{b}T}\rangle _{N}}

where $k_{B}$ is the Boltzmann constant, T is the temperature, N is the number of particles within the system 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 \Phi} is the interaction potential energy between the randomly placed test particle and the N particles that the system is comprised of.

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)