# Widom test-particle method

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 , 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

where is the Boltzmann constant and *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)