Test of the equipartition principle for granular spheres in a saw-tooth shaker

Abstract

The equipartition-of-energy principle has been tested for a simple model system of weightless granular media comprising inelastic spheres being shaken at a steady state in a uniaxially vibrating box. Granular dynamics simulations are reported and compared with a theoretical model based on the axioms of classical kinetic theory. For inelastic particles, both frictionless and with rotational degrees of freedom, the shaker reaches a steady state. Both the theoretical model and the simulation results show that equipartition always prevails at sufficiently small vibration amplitudes, irrespective of the frequency, over the whole density range. The independence of the reduced energy on frequency is an exact scaling result. When the shaker amplitude increases to the same order as a density-dependent characteristic path length, there are significant deviations from equipartition behaviour. In all the simulations, a Maxwell - Boltzmann distribution of velocities closely prevails within each degree of freedom, even when equipartition is not obtained between the longitudinal and transverse directions. The steady-state energetics of the saw-tooth shaker are evaluated analytically by means of an energy balance using exact kinetic theory and the known hard-sphere fluid collision frequency via the equation of state. This gives rise to scaling laws which enable granular-`thermodynamic' and transport properties of the fluidized granular material to be determined from the corresponding-state thermodynamic and transport properties of the classical hard-sphere fluid in thermal equilibrium.