content_image/416020.jpg content_image/416029.jpg content_image/416030.jpg content_image/416019.jpg
To probe the nature of the transition at low Γ , we plot in Fig. 5, τ = γ$_{max}$/γ versus ξ = ( Γ − Γ$_{c}$ ) /Γ$_{c}$ . The parameter τ can be thought as a characteristic re- laxation time for the evolution from a symmetric to an asymmetric configuration. The critical value Γ$_{c}$ of the adimensional acceleration was taken as the mean be- tween the highest Γ before γ becomes non zero, and the lowest value of Γ for which γ is different from zero. At the lower amplitudes it was observed that the critical value Γ$_{c}$ for the transition from zero growth to nonzero γ , shifted to higher values of Γ as the frequency in- creased. We can see how τ diverges as Γ approaches the critical value Γ$_{c}$ . More experiments have to be done before we can determine the nature of this transition. 

<img src='content_image/416020.jpg'>

Finally, we want to concentrate on the region of nonzero growth, for each curve we fixed the frequency and changed the amplitude A of vibration, assuming that γ$_{min}$ ≈ 0, we can rewrite equation (1) in the fol- lowing form: 

<img src='content_image/416029.jpg'>

where γ$_{n}$ = γ/γ$_{max}$ . 

This expression gives the value of the normalized growth rate as a function of Γ . For each frequency equation (2) is a solution of the following differential equation: 

<img src='content_image/416030.jpg'>

This equation is analogous to the well know logis- tic equation so we can picture γ$_{n}$ = 1 as a stable fixed point for γ$_{n}$ ( Γ ). Therefore as Γ increases the growth rate approaches its maximum value (which correspond 

to the stable fixed point). The smaller 1 /ΔΓ , the more energy needs to be supplied through vibration to reach the plateau of the curve γ$_{n}$ ( Γ ). In Fig. 6 we plotted the width ΔΓ of the region where γ grows. We observe that ΔΓ rises with the frequency up to f = 12 . 5 Hz and then drops almost an order of magnitude. This be- havior signals a qualitative change around f = 12 . 5 Hz . for this frequency 1 /ΔΓ is much smaller than for the other ones we measured, therefore around this partic- ular frequency we need to supply more energy to the granular bed to make it go from zero growth rate to the fastest growth. 

<img src='content_image/416019.jpg'>

A physical model that would explain our data and the above relationship for the growth rate and the adi- mensional acceleration for a given frequency is beyond the scope of this paper, but we consider that the sim- plicity of the results should help to suggest the rele- vant conditions that need to be satisfied by a reasonable model for the behavior of a vertically shaken granular bed in a partially filled U-tube. 

## 4 Conclusions 

We have made an experimental characterization of the rate of growth for the height difference Δh between the free levels of the granular columns in a partially filled U-tube subject to vertical vibrations. We have observed a continuous transition from zero growth to non zero growth rate γ . The region of finite growth was well described by a differential equation analogous to the logistic equation. We also observed an abrupt 

