Widom insertion method  The Widom insertion method is a statistical thermodynamic approach to the calculation of material and mixture properties. It is named for Benjamin Widom, who derived it in 1963. In general, there are two theoretical approaches to determining the statistical mechanical properties of materials. The first is the direct calculation of the overall partition function of the system, which directly yields the system free energy. The second approach, known as the Widom insertion method, instead derives from calculations centering on one molecule. The Widom insertion method directly yields the chemical potential of one component rather than the system free energy. This approach is most widely applied in molecular computer simulations but has also been applied in the development of analytical statistical mechanical models. The Widom insertion method can be understood as an application of the Jarzynski equality since it measures the excess free energy difference via the average work needed to perform, when changing the system from a state with N molecules to a state with N+1 molecules. Therefore it measures the excess chemical potential since formula_1, where formula_2. As originally formulated by Benjamin Widom in 1963, the approach can be summarized by the equation:  formula_3  where formula_4 is called the "insertion parameter", formula_5 is the number density of species formula_6, formula_7 is the activity of species formula_6, formula_9 is the Boltzmann constant, and formula_10 is temperature, and formula_11 is the interaction energy of an inserted particle with all other particles in the system. The average is over all possible insertions. This can be understood conceptually as fixing the location of all molecules in the system and then inserting a particle of species formula_6 at all locations through the system, averaging over a Boltzmann factor in its interaction energy over all of those locations. Note that in other ensembles like for example in the semi-grand canonical ensemble the Widom insertion method works with modified formulas. From the above equation and from the definition of activity, the insertion parameter may be related to the chemical potential by  formula_13  The pressure-temperature-density relation, or equation of state of a mixture is related to the insertion parameter via  formula_14  where formula_15 is the compressibility factor, formula_16 is the overall number density of the mixture, and formula_17 is a mole-fraction weighted average over all mixture components:  formula_18 formula_19  In the case of a 'hard core' repulsive model in which each molecule or atom consists of a hard core with an infinite repulsive potential, insertions in which two molecules occupy the same space will not contribute to the average.