Exact pair distribution function and structure factor for a one-dimensional hard rod mixture

A one-dimensional system of N hard rods with arbitary diameters and masses is considered. For this system the pair distribution function is calculated exactly. The result is independent of the masses but depends on the specific sequence of diameters that is considered. The pair distribution function is then specialized to a mixture of Nd particles with diameter d and ND particles with diameter D, respectively, and averaged over all possible arrangements of the particles. From this the structure factor S(κ) for finite N and it’s thermodynamic limit are obtained. It is shown how in this limit a finite peak tends to the singular contribution δ(κ). Finally, the limit S(κ→0) is calculated and compared with a formula following from general considerations for three-dimensional systems.