Scaling Properties of the Number of Random Sequential Adsorption Iterations Needed to Generate Saturated Random Packing
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Abstract
The properties of the number of iterations in random sequential adsorption protocol needed to generate finite saturated random packing of spherically symmetric shapes were studied. Numerical results obtained for one, two, and three dimensional packings were supported by analytical calculations valid for any dimension d. It has been shown that the number of iterations needed to generate finite saturated packing is subject to Pareto distribution with exponent \(11/d\) and the median of this distribution scales with packing size according to the powerlaw characterized by exponent d. Obtained results can be used in designing effective random sequential adsorption simulations.
Keywords
Random sequential adsorption Saturated random packings Spheres packings Multidimensional packings1 Introduction

a virtual particle is created. Its position and orientation within a packing is selected randomly.

the virtual particle is tested for overlaps with any of the other particles in the packing. If no overlap is found, it is added to the packing and holds its position and orientation until the end of simulations. Otherwise, the virtual particle is removed from the packing and abandoned.
The time t equals to the number of attempts of adding a particle per unit length for one dimensional packings, area for two dimensional packings, volume for three dimensional packings, etc.
The relation (1) was tested numerically to be valid for large enough, but finite packings [1, 14, 15, 16]. It is commonly used to estimate the number of particles at jamming, however it does not give any hints related to the number of RSA iterations needed to saturate a packing.
Recently, Zhang et al. have improved RSA algorithm, and showed that it is possible to generate saturated random packings of spherically symmetric particles in a reasonable simulation time [17]. The algorithm is based on tracking the area where placing subsequent particles is possible. The idea comes from earlier works concerning RSA on discrete lattices [18] and deposition of oriented squares on a continuous surface [19]. When particle is added to the packing this available area decreases. When it vanishes completely the packing is saturated. Because a single sampling of a particle centre from the available area of size s is equivalent to S / s samplings from the whole packing, this method makes it possible to determine number of RSA iterations in original protocol needed to saturate finite packing.
The aim of this paper is to analyse properties of this number as a random variable. In particular, it is interesting to investigate how it depends on packing size, since a recent study suggests that its median scales according to the powerlaw with an exponent equal to packing dimension [20].
2 Details of Numerical Simulations
Saturated random packing were generated using method described in detail in [17]. Line segments, disks and spheres, were packed onto a large line segment, square, and cube, respectively. In all these cases, periodic boundary conditions were used. The size (length, area or volume depending on a packing dimension) of packings varied from \(10^4\) to \(4 \times 10^7\) and a single shape had a unit volume. For each particular setup at least 100 of independent packings were created. For each of the saturated packings, the value of t was recorded at the moment when the last particle was added.
3 Results and Discussion
3.1 Distribution of Simulation Time
3.2 Median of Simulation Time
Numerical confirmation of this result is shown in Fig. 3. It is rather obvious that generating larger packings needs more computational time. But, the plot also shows that RSA in higher dimensions may be extremely time consuming as the time needed to generate a packing of the same size grows with dimension, as well as it scales with packing size with higher exponent. Moreover, packing fraction decreases with the growth of packing dimension [14, 17], which additionally spoils statistics in numerical simulations. Additionally, it is worth noting that Feder’s law (1) seems to be valid also for fractional packing’s dimensions [15, 16]. As its derivation bases on the same assumption as made here, namely (2), presented results should be valid also for fractional d’s. At last, the relation (1) works also for anisotropic shapes [21, 22], with parameter d denoting a number of degrees of freedom of adsorbed particle instead of packing dimension [16]. Therefore, the question arises if relations (8) and (11) are valid for anisotropic particles. But to answer it, the algorithm that generates saturated random packing in reasonable computational time is needed.
4 Conclusions
The number of RSA iterations needed to generate finite saturated random packing of spherically symmetric objects seems to subject to Pareto distribution with an exponent \((1+1/d)\) where d is packing dimension. The median of this distribution scales with packing size according to the powerlaw characterized by exponent d. Obtained results can be used to optimize and control time complexity of RSA simulations. In particular, together with results described in [20], they allow to design an RSA simulation that gives possibly the most accurate results within existing time limits.
Notes
Acknowledgements
I would like to thank Grzegorz Paja̧k for inspiring and fruitful discussions and Robert M. Ziff for comments on the paper. Part of numerical simulations was carried out with the support of the Interdisciplinary Centre for Mathematical and Computational Modeling (ICM) at University of Warsaw under Grant No. G278.
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