# Consistent particle systems and duality

## TL;DR

This paper explores consistent particle systems, revealing their equivalence to self-duality in reversible cases and deriving recursive relations for factorial moments, which aid in analyzing non-equilibrium steady states.

## Contribution

It establishes the equivalence between consistency and self-duality in reversible systems and derives recursive equations for factorial moments, advancing understanding of non-equilibrium dynamics.

## Key findings

- Consistency is equivalent to self-duality in reversible systems.
- Recursive equations link factorial moments of systems with n and n-1 particles.
- Universal recurrence relations for absorption probabilities and correlations in non-equilibrium states.

## Abstract

We consider consistent particle systems, which include independent random walkers, the symmetric exclusion and inclusion processes, as well as the dual of the KMP model. Consistent systems are such that the distribution obtained by first evolving $n$ particles and then removing a particle at random is the same as the one given by a random removal of a particle at the initial time followed by evolution of the remaining $n-1$ particles. In this paper we discuss two main results. Firstly, we show that, for reversible systems, the property of consistency is equivalent to self-duality, thus obtaining a novel probabilistic interpretation of the self-duality property. Secondly, we show that consistent particle systems satisfy a set of recursive equations. This recursions implies that factorial moments of a system with $n$ particles are linked to those of a system with $n-1$ particles, thus providing substantial information to study the dynamics. In particular, for a consistent system with absorption, the particle absorption probabilities satisfy universal recurrence relations. Since particle systems with absorption are often dual to boundary-driven non-equilibrium systems, the consistency property implies recurrence relations for expectations of correlations in non-equilibrium steady states. We illustrate these relations with several examples.