Experimental study on the vertical motion of grains in a vibrated U-tube

We investigate experimentally the behavior of the exponential growth rate of a column of grains, in a partially filled vertically shaken U-tube. For the set of frequencies used we identify three qualitatively different behaviors for the exponential growth rate γ as a function of the dimensionless acceleration Γ: (1) an interval of zero growth for low Γ with a smooth change to nonzero growth, analogous to a continuous phase transition; (2) a sigmoidal region for γ at intermediate values of the dimensionless acceleration Γ; and (3) an abrupt change from high values of γ to zero growth at high values of Γ, similar to a first order phase transition. We obtain that our experimental data is well described by a Boltzmann Sigmoidal funtion, for the change of the exponential growth rate with the dimensionless acceleration of the vertical vibrations.