Abstract
Several approximate forms of the three-particle distribution function are studied using error functions introduced to estimate the accuracy with which they satisfy the sequential relation connecting the two- and three-particle distribution functions. The procedure is based on modification and extension of the formal density-cluster expansion developed by Abe in the context of the classical statistical mechanics of an imperfect gas. Numerical evaluation of the error functions are carried out by expanding functions of the complicated multidimensional cluster integrals in terms of the associated Legendre functions. The results obtained numerically in the first three lowest-order approximations for the ground state of liquid $^{4}\mathrm{He}$ at equilibrium density show that the overall magnitude of the error function decreases significantly as the order of approximation increases to the next higher order.
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