Generalizations of the Virial and Wall Theorems in Classical Statistical Mechanics

A generalized virial theorem which expresses inverse compressibility in terms of integrals of virials and canonical distribution functions through the four-particle distribution is transformed to the grand canonical ensemble and becomes an expression for compressibility in terms of the same integrals formed with grand canonical distribution functions. The integrals are of a mixed (virial and fluctuation) type. While the thermodynamic functions expressed by the same integrals with canonical and grand canonical distribution functions are quite different, the two formulas agree in the thermodynamic limit because of the different asymptotic behavior of canonical and grand canonical distribution functions. We also derived an alternative form of the second virial theorem which expresses compressibility in terms of integrals over virials and grand canonical distribution functions through the three-particle distribution function only. It is shown that this form and its generalizations to higher derivatives of the density, as well as the hierarchy of fluctuation theorems and the fugacity expansions of distribution and correlation functions can all be very simply derived from a set of integro-differential equations satisfied by the grand canonical distribution functions. A generalization of the wall theorem (P/kT = ρwall) is derived and shown to be equivalent to the generalized virial theorem (canonical form).