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tokens: the effect of granular convection. Small coloured particles were put on the bottom layer, and the entire apparatus was shaken at Γ = 4.0 for around 12 000 cycles. After the oscillations, it was confirmed that the coloured small particles either remained at the bottom or were pushed to the cylinder wall, but none of them moved to the centre and rose upwards. This shows that granular convection was only slightly induced beside the cylinder wall, but it was not the dominant mechanism for the BNE in our experimental set-up. In this sense, both our numerical simulations and hardware simulations suggest the existence of ‘less-convective’ conditions. 4.2 Results and discussion Variation of the vibration amplitude The necessary conditions to induce the BNE were first investigated by focusing on the vibration amplitude. The amplitude was varied between 2.5, 5.0, and 10.0 mm, and the frequency was also varied to vary the parameter Γ for each amplitude. Fig. 12 shows the results of the evaluation of the BNE and the RBNE. For each amplitude value, the RBNE was induced over Γ ≈ 3.5, which is consistent with the evaluation by the one-dimensional model. Fig. 13 (upper panel) shows the number of cycles it took for the intruder to rise to the surface. The numbers of cycles required to rise to the surface was much larger than for the nominal cases of the simulations as a whole, especially under the conditions of case (b) for the one-dimensional model. This is because of the size difference of the small particles or the fact that both the particle–particle and particle–wall friction constants were lower in the laboratory experiments. It is clear that the BNE is more likely to occur as Γ increases. The decrease in the number of cycles plateaued at Γ ≈ 1.9 and this indicates that the mechanism for inducing the BNE changed at this point. This result is interpreted as the boundary between the cases (a) and (b) for the one-dimensional model with the coefficient of restitution being 0.4 e 0.6, which was estimated from Fig. 3 . This is also fairly consistent with the assumptions of the numerical simulations. The estimated coefficient of restitution is an ‘empirical’ value, which may be different from the value obtained in the bouncing experiments. It can be stated that the cases (a) and (b) are, respectively, the ‘slow BNE’ case and the ‘fast BNE’ case. For the cases categorized as (a) in the one-dimensional model where Γ 1.9, it was evaluated by Δ h = Af (Γ, e ) (Fig. 2 ) as is shown in Fig. 13 (lower panel), where e = 0.6. Even if the amplitude was different, the number of cycles was similar for the same value of Δ h , which is compatible with our hypothesis. Variation of the density of the intruder The density of the intruder was varied and the vibration amplitude was fixed at 5.0 mm. Three kinds of material were prepared for the intruder: acryl (1.2 g cm −3 ), glass (2.4 g cm −3 ), and high-carbon chromium steel (7.8 g cm −3 ), which were all identical to the