Widom test-particle method

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The Widom test-particle method, proposed by Benjamin Widom [1][2], is 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^{excess} = \mu -\mu^{ideal}  = -k_BT \log \langle e^{-\Delta\Phi/k_BT}\rangle_N ,

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

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