Glassy behavior of the parking lot model

Abstract

We present a theoretical discussion of the reversible parking problem, which appears to be one of the simplest systems exhibiting glassy behavior. The existence of slow relaxation, nontrivial fluctuations, and an annealing effect can all be understood by recognizing that two different time scales are present in the problem. One of these scales corresponds to the fast filling of existing voids, the other is associated with collective processes that overcome partial ergodicity breaking.The results of the theory are in a good agreement with simulation 1 data; they provide a simple qualitative picture for understanding recent granular compaction experiments and other glassy systems.

Adsorption-desorption model and its application to vibrated granular materials

“…which reduces to a power law, 1/t, when ln(K) << t << K. This result is equivalent to the ln(ω)-behavior already predicted along similar lines similar by Kolan et al [19]. The insets in Figs.…”

“…2b, the relaxation rate is plotted as a function of K: the dashed curve gives the mean-field prediction, Eq. (19), and open circles correspond to the best exponential fit to the simulation results. It is evident that the mean-field analysis gives a poor estimate of the relaxation rate for large K. This failure can be understood by the noting that the mean-field assumption leads to a characteristic time for the rearrangement of Φ of the order K/ ln(K) 2 , i.e., much smaller than K, the characteristic time for desorption.…”

Statistical mechanical description of the parking-lot model for vibrated granular materials

“…Except for the two above mentioned limits (RSA when k − = 0, equilibrium when t → +∞), the infinite hierarchy of coupled equations cannot be solved analytically and one must resort to approximate treatments and computer simulations, as described in previous articles [37,38,39,40]. First introduced in the context of protein adsorption at liquid-solid interfaces [34,35,36], the random adsorptiondesorption model has recently been applied to the description of weakly vibrated granular materials [6,7,37,38,39,40,41]. The connection between the parking-lot model and these latter is made by regarding the particles on the line as an average layer of grains in the vibrated column.…”

Reversible random sequential adsorption of dimers on a triangular lattice

“…1 confirms the mean field prediction for the equilibrium values of ρ. However, as with the continuous parking lot model [14], the mean-field description is unable to accurately predict the time evolution of ρ. This can be seen in Fig.…”