Multibody Interactions in Coarse-Graining Schemes for Extended Systems

In [12, 14] coarse-graining schemes for stochastic many body microscopic models focusing on equilibrium stochastic lattice systems were developed and furthermore, the accuracy of the coarse-graining schemes was estimated in terms of information loss between the exact and approximate coarse-grained Gibbs measures. The relative entropy estimates are given in terms of a small parameter . In this paper we focus on the error quantification of the coarse-graine approximation and identifying the parameter based on the dependence on the underlying potential. In particular we study the role of multi-body interactions which are often not included in coarsegraining schemes as they can be computational expensive. The derived error estimates suggests strategies to overcome the computational expense due to the inclusion of multi-body interactions by either (a) truncating the multi-body interactions or by (b) splitting the potential into a short range piece where multi-body interactions are necessary and a smooth long range component where twobody interactions are sufficient. We present numerical examples that demonstrate that the inclusion of multi-body interactions shows substantial improvement in the dynamics even in regimes where the effect of multi-body interactions in predicting equilibrium properites is minimal, see [14]. We conclude with a detailed discussion on the relevance and impact of the multi-body interactions in off-lattice sytems.