# Effects of boundary conditions on irreversible dynamics

## TL;DR

This paper investigates how boundary conditions influence the stationary states of a one-dimensional Ising spin system with asymmetric dynamics at low temperature, revealing significant effects of boundary modifications.

## Contribution

It introduces a series expansion approach to analyze the impact of boundary conditions on the stationary measure of an asymmetric Ising system at low temperature.

## Key findings

- Boundary conditions drastically alter the stationary measure.
- Series expansion method effectively captures boundary effects.
- Macroscopic changes occur due to single-site boundary modifications.

## Abstract

We present a simple one-dimensional Ising-type spin system on which we define a completely asymmetric Markovian single spin-flip dynamics. We study the system at a very low, yet non-zero, temperature and we show that for empty boundary conditions the Gibbs measure is stationary for such dynamics, while introducing in a single site a $+$ condition the stationary measure changes drastically, with macroscopical effects. We achieve this result defining an absolutely convergent series expansion of the stationary measure around the zero temperature system. Interesting combinatorial identities are involved in the proofs.