The primary objective of this work is to develop coarse-grainingschemes for stochastic many-body microscopic models and quantify theireffectiveness in terms of a priori and a posteriori error analysis. Inthis paper we focus on stochastic lattice systems ofinteracting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grainedapproximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first – and often inadequate – approximation and obtain more accurate coarse-graining schemes.The cluster expansions yield also sharp a posteriori error estimates forthe coarse-grained approximations that can be used for the construction ofadaptive coarse-graining methods.
We present a number of numerical examples that demonstrate that thecoarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improvepredictions of earlier coarse-graining schemes for short-range interactions.