Statistical mechanics of vibration-induced compaction of powders

We propose a theory that describes the density relaxation of loosely packed, cohesionless granular material under mechanical tapping. Using the compactivity concept, we develop a formalism of statistical mechanics that allows us to calculate the density of a powder as a function of time and compactivity. A simple fluctuation-dissipation relation that relates compactivity to the amplitude and frequency of a tapping is proposed. The experimental data of Nowak et al. [Powder Technol. 94, 79 (1997)] show how the density of powder initially deposited in a fluffy state evolves under carefully controlled tapping towards a random close packing (RCP) density. Ramping the vibration amplitude repeatedly up and back down again reveals the existence of reversible and irreversible branches in the response. In the framework of our approach the reversible branch (along which the RCP density is obtained) corresponds to the steady-state solution of the Fokker-Planck equation, whereas the irreversible one is represented by a superposition of ``excited state'' eigenfunctions. These two regimes of response are analyzed theoretically and a qualitative explanation of the hysteresis curve is offered.