Lattice models for ballistic aggregation in one-dimension

TL;DR

This paper introduces two one-dimensional lattice models with stochastic particle hopping and aggregation, which replicate the long-term behavior and velocity distribution of continuum sticky gas dynamics, validated through numerical comparison.

Contribution

The authors develop lattice models that accurately mimic continuum sticky gas behavior, including velocity distribution and shock profiles, bridging discrete and continuum approaches.

Findings

Lattice models match MD results for sticky gas behavior.

Non-Gaussian velocity distribution tail is reproduced.

Spatial velocity profiles with shocks are strikingly similar.

Abstract

We propose two lattice models in one dimension, with stochastically hopping particles which aggregate on contact. The hops are guided by "velocity rates" which themselves evolve according to the rules of ballistic aggregation as in a sticky gas in continuum. Our lattice models have both velocity and density fields and an appropriate real time evolution, such that they can be compared directly with event driven molecular dynamics (MD) results for the sticky gas. We demonstrate numerically that the long time and large distance behavior of the lattice models are identical to that of the MD, and some exact results known for the sticky gas. In particular, the exactly predicted form of the non-Gaussian tail of the velocity distribution function is clearly exhibited. This correspondence of the lattice models and the sticky gas in continuum is nontrivial, as the latter has a deterministic dynamics with local kinematic constraint, in contrast with the former; yet the spatial velocity profiles (with shocks) of the lattice models and the MD have striking match.