Bulk distribution functions and fluctuation theorems

A new integral relation linking the bulk distribution functions of a one phase fluid is given. It is related to the Ornstein--Zernike compressibility formula, which is also discussed and generalized. In addition, an expression is found for the probability that a region of volume V in an infinite system of given density and temperature contains exactly N particles. This enables the fluctuation--compressibility relation and its generalization to be calculated, without the use of the grand canonical ensemble