Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems

TL;DR

This paper develops and analyzes coarse-graining schemes for stochastic lattice systems, providing error estimates and adaptive methods to improve predictions of critical phenomena and phase transitions in many-body models.

Contribution

It introduces new coarse-graining algorithms with a posteriori error estimates based on relative entropy and cluster expansions for stochastic lattice systems.

Findings

Accurate predictions of critical behavior and hysteresis achieved.

Cluster expansion improves coarse-graining accuracy for long-range interactions.

Error estimates enable adaptive coarse-graining methods.

Abstract

The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. %such as Ising-type models. The proposed algorithms are derived from an initial coarse-grained approximation 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 for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-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 improve predictions of earlier coarse-graining schemes for short-range interactions.