This similarity is not accidental; indeed, substituting (12) in the relations above for the thermodynamic parameters (Equations 7, 9 and 10) yields the corresponding virial expansions. The auxiliary function y ( r ) is known as the cavity distribution function. It has been shown that for classical fluids at a fixed density and a fixed positive temperature, the effective pair potential that generates a given g ( r ) under equilibrium is unique up to an additive constant, if it exists.