Exact solution for the singlet density distributions and second-order correlations of normal-mode coordinates for hard rods in one dimension

We examine the distribution of normal-mode coordinates (defined via the eigenvectors of a chain of harmonic oscillators) for a system of purely repulsive hard rods in one dimension. We obtain an exact solution for the singlet density distribution, and separately for the covariances of the normal-mode coordinates. The hard-rod behavior is examined in terms of its deviation from the corresponding distributions for the system of harmonic oscillators. All off-diagonal covariances are zero in the hard-rod system, and the (on-diagonal) variances vary with the normal-mode wave number exactly as in the harmonic system. The detailed singlet normal-mode density distributions are very smooth but nonanalytic, and they differ from the (Gaussian) distributions of the corresponding harmonic system. However, all of the normal-mode coordinate distributions differ in roughly the same way when properly scaled by the distribution variance, and the differences vanish as 1/N in the thermodynamic limit of an infinite number of particles N.