Energy flows in vibrated granular media

We study vibrated granular media, investigating each of the three components of the energy flow: particle-particle dissipation, energy input at the vibrating wall, and particle-wall dissipation. Energy dissipated by interparticle collisions is well estimated by existing theories when the granular material is dilute, and these theories are extended to include rotational kinetic energy. When the granular material is dense, the observed particle-particle dissipation rate decreases to as little as $2/5$ of the theoretical prediction. We observe that the rate of energy input is the weight of the granular material times an average vibration velocity times a function of the ratio of particle to vibration velocity. ``Particle-wall'' dissipation has been neglected in all theories up to now, but can play an important role when the granular material is dilute. The ratio between gravitational potential energy and kinetic energy can vary by as much as a factor of 3. Previous simulations and experiments have shown that $E\ensuremath{\propto}{V}^{\ensuremath{\delta}}$, with $\ensuremath{\delta}=2$ for dilute granular material, and $\ensuremath{\delta}\ensuremath{\approx}1.5$ for dense granular material. We relate this change in exponent to the departure of particle-particle dissipation from its theoretical value.