# Optimal random deposition of interacting particles

## TL;DR

This paper derives exact solutions for the kinetics of interacting particle deposition in one dimension, revealing how to optimize packing density through interaction potentials, with implications for biological systems like DNA nucleosome arrangement.

## Contribution

It provides the first exact analytical solution for one-dimensional interacting particle deposition with finite-range potentials, identifying a unique optimal potential for maximum density.

## Key findings

- Exact time-dependent solutions for deposition processes
- Existence of a unique optimal interaction potential
- Optimal packing requires highly coordinated dynamics

## Abstract

Irreversible random sequential deposition of interacting particles is widely used to model aggregation phenomena in physical, chemical, and biophysical systems. We show that in one dimension the exact time dependent solution of such processes can be found for arbitrary interaction potentials with finite range. The exact solution allows to rigorously prove characteristic features of the deposition kinetics, which have previously only been accessible by simulations. We show in particular that a unique interaction potential exists that leads to a maximally dense line coverage for a given interaction range. Remarkably, this distribution is singular and can only be expressed as a mathematical limit. The relevance of these results for models of nucleosome packing on DNA is discussed. The results highlight how the generation of an optimally dense packing requires a highly coordinated packing dynamics, which can be effectively tuned by the interaction potential even in the presence of intrinsic randomness.