# Evaluation on asymptotic distribution of particle systems expressed by probabilistic cellular automata

## TL;DR

This paper investigates the asymptotic behavior of probabilistic cellular automata modeling particle systems, proposing conjectures and deriving formulas for their steady states and relations between density and flux.

## Contribution

It introduces conjectures for the asymptotic distribution of PBCA and derives explicit formulas involving hypergeometric functions, extending analysis to infinite space.

## Key findings

- Asymptotic distribution converges to a unique steady state.
- Derived relations between particle density and flux in the infinite limit.
- Proposed extended systems with analyzable asymptotic behavior.

## Abstract

We propose some conjectures for asymptotic distribution of probabilistic Burgers cellular automaton (PBCA) which is defined by a simple motion rule of particles including a probabilistic parameter. Asymptotic distribution of configurations converges to a unique steady state for PBCA. We assume some conjecture on the distribution and derive the asymptotic probability expressed by GKZ hypergeometric function. If we take a limit of space size to infinity, a relation between density and flux of particles for infinite space size can be evaluated. Moreover, we propose two extended systems of PBCA of which asymptotic behavior can be analyzed as PBCA.