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 \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 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

\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.


  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)