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Multidisperse Random Sequential Adsorption and Generalizations

Roger Fan, Nitya Mani

Abstract

In this paper, we present a unified study of the limiting density in one-dimensional random sequential adsorption (RSA) processes where segment lengths are drawn from a given distribution. In addition to generic bounds, we are also able to characterize specific cases, including multidisperse RSA, in which we draw from a finite set of lengths, and power-law RSA, in which we draw lengths from a power-law distribution.

Multidisperse Random Sequential Adsorption and Generalizations

Abstract

In this paper, we present a unified study of the limiting density in one-dimensional random sequential adsorption (RSA) processes where segment lengths are drawn from a given distribution. In addition to generic bounds, we are also able to characterize specific cases, including multidisperse RSA, in which we draw from a finite set of lengths, and power-law RSA, in which we draw lengths from a power-law distribution.
Paper Structure (14 sections, 26 theorems, 129 equations, 6 figures)

This paper contains 14 sections, 26 theorems, 129 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Outline
  5. Preliminaries
  6. Notation
  7. The $\nu$-RSA process
  8. Analytic Tools
  9. Warmup: The Multidisperse Ghost Process
  10. The Multidisperse RSA Process
  11. General Length Distributions
  12. Convergent Length Distributions
  13. Divergent Length Distributions
  14. Power Function Distribution

Key Result

Theorem 1.1

Let $N_L$ be defined as above. Then, where $\alpha$ is the Rényi parking constant, given by

Figures (6)

  • Figure 1: Rényi's parking problem.
  • Figure 2: Different RSA processes.
  • Figure 3: Growth of $\mathop{\mathrm{\mathbb{E}}}\nolimits[S_L]$ with various length distributions.
  • Figure 4: Plots of $\mathop{\mathrm{\mathbb{E}}}\nolimits[N_{k,L}]$ in $\mathop{\mathrm{\mathcal{M}}}\nolimits ( (1,.5), (1.3, .3), (1.5, .2) )$.
  • Figure 5: $\mathop{\mathrm{\mathbb{E}}}\nolimits[S_L]$ grows linearly under various convergent ldfs.
  • ...and 1 more figures

Theorems & Definitions (69)

  • Theorem 1.1: Rényi
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: The $\nu$-RSA process
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7: Laplace Transform Properties
  • Theorem 2.8: Hardy-Littlewood Tauberian Theorem
  • Proposition 2.9
  • ...and 59 more