Asymptotic Value of the Pair Distribution Near a Wall

Abstract

We calculate the asymptotic value of the pair probability density ρ₂(r₂, r₁) for finding a fluid particle at a point r₂ far in the interior of a fluid, when it is known that there is a particle at r₁ in contact with the walls (rigid) of the container. This value is different from the well‐known expression for the asymptotic value of ρ₂(r₂, r₁) when both r₂ and r₁ are in the interior of the fluid. Our derivation is based on the virial theorem for total momentum fluctuations in an equilibrium system and makes use of the assumption that there are no long range correlations in a fluid. Application is made of our result to re‐derive simply the expression for the second virial coefficient and the exact equation of state of a hard‐sphere gas in one dimension. Quantum systems are also treated.