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We consider consistent particle systems, which include independent randomwalkers, the symmetric exclusion and inclusion processes, as well as the dualof the KMP model. Consistent systems are such that the distribution obtained byfirst evolving $n$ particles and then removing a particle at random is the sameas the one given by a random removal of a particle at the initial time followedby evolution of the remaining $n-1$ particles. In this paper we discuss twomain results. Firstly, we show that, for reversible systems, the property ofconsistency is equivalent to self-duality, thus obtaining a novel probabilisticinterpretation of the self-duality property. Secondly, we show that consistentparticle systems satisfy a set of recursive equations. This recursions impliesthat factorial moments of a system with $n$ particles are linked to those of asystem with $n-1$ particles, thus providing substantial information to studythe dynamics. In particular, for a consistent system with absorption, theparticle absorption probabilities satisfy universal recurrence relations. Sinceparticle systems with absorption are often dual to boundary-drivennon-equilibrium systems, the consistency property implies recurrence relationsfor expectations of correlations in non-equilibrium steady states. Weillustrate these relations with several examples.

Consistent particle systems and duality