Dynamics of the breakdown of granular clusters

Abstract

Recently van der Meer et al. studied the breakdown of a granular cluster [Phys. Rev. Lett. 88, 174302 (2002)]. We reexamine this problem using an urn model, which takes into account fluctuations and finite-size effects. General arguments are given for the absence of a continuous transition when the number of urns (compartments) is greater than two. Monte Carlo simulations show that the lifetime of a cluster tau diverges at the limits of stability as tau approximately N(1/3), where N is the number of balls. After the breakdown, depending on the dynamical rules of our urn model, either normal or anomalous diffusion of the cluster takes place.

Analytic study of the three-urn model for separation of sand

“…The urn model proposed in Ref. [5] displays the same phenomenon: In region I a granular cluster is stable up to a time scale τ , after which sand particles are distributed uniformly over all boxes. Approaching the phase boundary I-IV, the characteristic time scale diverges.…”

“…We analytically investigate the three-urn model introduced by Coppex et al [5] We formally solve the master equation of the model in the thermodynamic limit and find how the probability distribution evolves. In the long time limit, the probability distribution becomes delta peaks only at the stable fixed points.…”

“…Approaching the phase boundary I-IV, the characteristic time scale diverges. At the phase boundary, it is found numerically that τ scales as τ ∼ N z with z ≃ 0.32 [5]. In this section, the scaling law for the characteristic time τ at and near the phase boundary is derived analytically.…”

“…It was also found that the characteristic time which it takes to reach the symmetric state from an asymmetric state is given by the same free energy function which governs the stationary probability distribution [6]. Coppex et al also numerically investigated the three-urn model to find the absence of continuous transition [5]. They also obtained that the characteristic time at the limit of stability diverges algebraically as the number of particles increases (see also Ref.…”

Lotka-Volterra Model of Macro-Evolution on Dynamical Networks

“…One can also examine a variable number of species that would allow to create new species using certain mutation mechanism rather than assuming that they appear as soon as a niche becomes empty. Another possibility that is outside the scope of majority of macro-evolutionary models, and that will be discussed in the forthcoming paper [18], is to examine emergent properties of species.…”