Derivation of the formula to calculate the chemical potential difference for the components involved in identity exchange moves in Gibbs ensemble simulation of mixtures

Since temperature and pressure are specified at the beginning of a simulation run in the Gibbs ensemble Monte Carlo (GEMC) method for mixtures, the condition of equilibrium is fulfilled through two sets of equalities in each phase: one set for the chemical potentials of the components of smaller molecules (calculated from transfer trial moves), and the other set for the differences between the chemical potentials of the components of larger molecules and the components of smaller molecules (calculated from identity exchange trial moves). The formula to calculate the former quantities is known since the time the GEMC method was proposed. However, the formula to calculate the latter quantities has recently been given in the literature without a formal derivation. In this work, a statistical-mechanical derivation of that formula is presented, within the framework of the canonical ensemble, and some justification is given for its extension to the Gibbs ensemble.