Duality in interacting particle systems out of equilibrium

Abstract

In this thesis we study a class of interacting particle systems sharing a duality property. This class includes the Symmetric Inclusion Process (SIP(2k)), the Symmetric Exclusion Process (SEP(2j)) and the Independent Random Walkers (IRW). When these systems are in equilibrium (namely they are isolated from the exterior) they admit stationary measures that are also reversible product space-homogeneous measures. However when the systems are in contact with reservoirs that keep them out of equilibrium, the reversibility is lost and the stationary microscopic distributions are unknown with the exception for the SEP(1) (the Symmetric Exclusion Process) and the IRW. In order to examine the non-equilibrium stationary distributions for the whole class of processes, we make use of duality between these processes and particle systems with absorbing boundaries. Our first main results are in section 6, where we give explicit formulas for the absorption probabilities for the dual systems with 1 and 2 particles. Then we use this result to compute an explicit formula for the variance and covariance of the sites occupation numbers for the many-particles systems with density reservoirs in the non-equilibrium stationary distribution in section 7. Then we identify three possible scaling regimes for the density field: a deterministic regime where the variance vanishes and particles are expected to converge to independent Brownian motions; a sticky regime where the variance is finite and particles are expected to converge to sticky Brownian motions; and finally an absorbing regime where the variance is infinite and particles are expected to converge to coalescing Brownian motions. In the last part of the thesis (section 8) we start the analysis of the dynamics of the particle systems.