An analytic model for a cooperative ballistic deposition in one dimension

TL;DR

This paper introduces an analytic model for cooperative ballistic deposition in one dimension, exploring how attractive interactions influence packing density, cluster formation, and the approach to maximum coverage.

Contribution

It develops a tunable parameter model interpolating between classical parking, ballistic deposition, and cooperative deposition, providing analytical insights into the effects of attraction on packing and clustering.

Findings

Jamming coverage increases with attraction strength a

System approaches closest packing but does not percolate in 1D

Mean cluster size scales as a^{1/2} in large a regime

Abstract

We formulate a model for a cooperative ballistic deposition (CBD) process whereby the incoming particles are correlated with the ones already adsorbed via attractive force. The strength of the correlation is controlled by a tunable parameter $a$ that interpolates the classical car parking problem at $a=0$, the ballistic deposition at $a=1$ and the CBD model at $a>1$. The effects of the correlation in the CBD model are as follows. The jamming coverage $q(a)$ increases with the strength of attraction $a$ due to an ever increasing tendency of cluster formation. The system almost reaches the closest packing structure as $a\to\infty$ but never forms a percolating cluster which is typical to 1D system. In the large $a$ regime, the mean cluster size $k$ increases as $a^{1/2}$. Furthermore, the asymptotic approach towards the closest packing is purely algebraic both with $a$ as $q(\infty)-q(a) \sim a^{-1/2}$ and with $k$ as $q(\infty)-q(k) \sim k^{-1}$ where $q(\infty)\simeq 1$.