Equilibrium distributions of a three-dimensional hard-sphere system under gravity

For a hard-sphere system under gravity, confined in a closed cubic container in space, the speed and height distributions of the three-dimensional spherical particles are studied in detail for the state of thermodynamic equilibrium. The system parameters - the total amount of particles, the radius of spheres, the particles initial velocity distribution, the magnitude of the acceleration due to gravity - are varied in a wide range. These equilibrium distributions are obtained numerically by means of direct computer simulation of a many-particle system while it evolves from a given initial (non-equilibrium) state to equilibrium. It is found that when the particles occupy just approximately 3% of the whole volume of the container there is a deviation in the particles equilibrium height distribution from the Boltzmann one. When the particles occupy a larger part of the container an inflexion point occurs in the equilibrium height distribution. This means that for analytical description of such densities the granular system should be considered as non-ideal. We found a good agreement between the numerical simulation results and the density-functional description of the height distribution based on the van der Waals equation of state for hard spherical grains.