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We study conservative particle systems on W^S, where S is countable and W ={0, ..., N} or the natural numbers. The rate of a particle moving from site xto site y is given by p(x,y) b(eta_x, eta_y), where eta_z is the number ofparticles at site z. Under assumptions on b and the assumption that p is finiterange, which allow for the exclusion, zero range and misanthrope processes, weshow exactly what the stationary product measures are.  Furthermore we show that a stationary measure mu is ergodic if and only ifthe tail sigma algebra of the partial sums is trivial under mu. This is aconsequence of a more general result on interacting particle systems that showsthat a stationary measure is ergodic if and only if the sigma algebra of setsinvariant under the transformations of the process is trivial. We apply thisresult combined with a coupling argument on the stationary product measures todetermine which product measures are ergodic. For the case that W is finitethis gives a complete characterisation.  For the case that W is the set of natural numbers we show that for nearly allfunctions b a stationary product measure is ergodic if and only if it issupported by configurations with an infinite amount of particles. We show thatthis picture is not complete, we give an example of a system where b is suchthat there is a stationary product measure which is not ergodic, even though itconcentrates on configurations with an infinite number of particles.

Stationary product measures for conservative particle systems and ergodicity criteria