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interval Δγ (gives the range of growth rate from the lower asymptote to upper asymptote); (c) the inflec- tion point Γ$_{i}$ ; and (d) the width ΔΓ of the transition region. 

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A plot of γ$_{max}$ as a function of the frequency is shown in Fig. 4. This parameter is the saturation growth rate and gives an upper limit for the speed of the up- ward granular transport through the tube. We observe that the ability of the granular material to flow up- wards is reduced as the frequency increases. For higher frequencies we can expect that the granular system ap- proaches a liquid like behavior, and consequently, as the frequency rises it becomes more difficult for the granu- lar bed to climb up. 

For each frequency we can separate the observed be- havior in Fig. 3 in three regions. One for which γ = 0 at low Γ , the second one for which γ ̸ = 0, for intermediate values of Γ ; and a third region for which γ = 0 and Γ is high. We can speculate that, for a given frequency, at low Γ ; the granular bed behaves like a solid; at inter- mediate values of Γ ; a mixed state occurs were the solid like state and the fluid state combine in a complex way to produce the collective upward motion of the grains in one of the branches of the container; and for suffi- ciently high Γ the liquid like state dominates and the grains are unable to climb up on one of the branches of the tube. 

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We will now focus below on two types of transitions that were observed. One of them was a (seemingly con- tinuous) transition from zero growth at the lowest am- plitudes to a finite γ at higher amplitudes. The other was a “jump” in the value of γ (resembling a first order phase transition), which marked a transition from finite growth rate to no growth, when Γ was sufficiently large. A detailed study of each transition requires further in- vestigations. For the case of the abrupt transition, a different experimental setup capable of reaching larger amplitudes is necessary. Nevertheless, our data could give us an insight on the nature of the transition at low Γ . 

