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### Mathematical strategies in the coarse-graining of extensive systems : error quantification and adaptivity

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Katsoulakis, Markos A., Plechac, Petr, Rey-Bellet, Luc and Tsagkarogiannis, Dimitrios K.
(2008)
*Mathematical strategies in the coarse-graining of extensive systems : error quantification and adaptivity.*
In: 4th International Workshop on Equilibrium Thermodynamics and Complex Fluids, Rhodes, Greece, Sep 03-07, 2006. Published in: Journal of Non-Newtonian Fluid Mechanics, Vol.152
(No.1-3). pp. 101-112.

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1016/j.jnnfm.2007.05.005

## Abstract

In this paper we continue our study of coarse-graining schemes for stochastic many-body microscopic models started in Katsoulakis et al. [M. Katsoulakis, A. Majda, D. Vlachos, Coarse-grained stochastic processes for microscopic lattice systems, Proc. Natl. Acad. Sci. 100 (2003) 782-782, M.A. Katsoulakis, L. Rey-Bellet, P. Plechac, D. Tsagkarogiannis, Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems, M2AN Math. Model. Numer. Anal., in press], focusing on equilibrium stochastic lattice systems. Using cluster expansion techniques we expand the exact coarse-grained Hamiltonian around a first approximation and derive higher accuracy schemes by including more terms in the expansion. The accuracy of the coarse-graining schemes is measured in terms of information loss, i.e., relative entropy, between the exact and approximate coarse-grained Gibbs measures. We test the effectiveness of our schemes in systems with competing short- and long-range interactions, using an analytically solvable model as a computational benchmark. Furthermore, the cluster expansion in Katsoulakis et al. [M.A. Katsoulakis, L. Rey-Bellet, P. Plechac, D. Tsagkarogiannis, Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems, M2AN Math. Model. Numer. Anal., in press] yields sharp a posteriori error estimates for the coarse-grained approximations that can be computed on-the-fly during the simulation. Based on these estimates we develop a numerical strategy to assess the quality of the coarse-graining and suitably refine or coarsen the simulations. We demonstrate the use of this diagnostic tool in the numerical calculation of phase diagrams. (C) 2007 Elsevier B.V. All rights reserved.

Item Type: | Conference Item (UNSPECIFIED) |
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Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Error analysis (Mathematics), Entropy, Lattice theory, Monte Carlo method, Stochastic processes |

Journal or Publication Title: | Journal of Non-Newtonian Fluid Mechanics |

Publisher: | Elsevier BV |

ISSN: | 0377-0257 |

Official Date: | June 2008 |

Volume: | Vol.152 |

Number: | No.1-3 |

Number of Pages: | 12 |

Page Range: | pp. 101-112 |

Identification Number: | 10.1016/j.jnnfm.2007.05.005 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Deutsche Forschungsgemeinschaft (DFG), National Science Foundation (U.S.) (NSF) |

Grant number: | DE-FG02-05ER25702 (DFG), NSF-DMS-0413864 (NSF), NSF-ITR-0219211 (NSF), NSF-DMS-0303565 (NSF), NSF-DMS-0605058 (NSF) |

Title of Event: | 4th International Workshop on Equilibrium Thermodynamics and Complex Fluids |

Type of Event: | Workshop |

Location of Event: | Rhodes, Greece |

Date(s) of Event: | Sep 03-07, 2006 |

References: | [1] C.F. Abrams, K. Kremer, The effect of bond length on the structure of dense |

URI: | http://wrap.warwick.ac.uk/id/eprint/29832 |

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