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Coarse-graining schemes for stochastic lattice systems with short and long-range interactions


Authors: Markos A. Katsoulakis, Petr Plecháč, Luc Rey-Bellet and Dimitrios K. Tsagkarogiannis
Journal: Math. Comp. 83 (2014), 1757-1793
MSC (2010): Primary 65C05, 65C20, 82B20, 82B80, 82-08
Published electronically: March 25, 2014
MathSciNet review: 3194129
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Abstract: We develop coarse-graining schemes for stochastic many-particle microscopic models with competing short- and long-range interactions on a $ d$-dimensional lattice. We focus on the coarse-graining of equilibrium Gibbs states, and by using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical coarse-graining schemes that are accompanied by a posteriori error estimates for lattice systems with short- and long-range interactions.


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  • [1] Reinier L. C. Akkermans and W. J. Briels.
    Coarse-grained interactions in polymer melts: A variational approach.
    J. Chem. Phys., 115(13):6210-6219, 2001.
  • [2] Sasanka Are, Markos A. Katsoulakis, Petr Plecháč, and Luc Rey-Bellet, Multibody interactions in coarse-graining schemes for extended systems, SIAM J. Sci. Comput. 31 (2008/09), no. 2, 987–1015. MR 2466145, 10.1137/080713276
  • [3] Rodney J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989. Reprint of the 1982 original. MR 998375
  • [4] Lorenzo Bertini, Emilio N. M. Cirillo, and Enzo Olivieri, Renormalization-group transformations under strong mixing conditions: Gibbsianness and convergence of renormalized interactions, J. Statist. Phys. 97 (1999), no. 5-6, 831–915. MR 1734386, 10.1023/A:1004620929047
  • [5] Anton Bovier and Miloš Zahradník, A simple inductive approach to the problem of convergence of cluster expansions of polymer models, J. Statist. Phys. 100 (2000), no. 3-4, 765–778. MR 1788485, 10.1023/A:1018631710626
  • [6] M. Cassandro and E. Presutti, Phase transitions in Ising systems with long but finite range interactions, Markov Process. Related Fields 2 (1996), no. 2, 241–262. MR 1414119
  • [7] A. Chatterjee, M. Katsoulakis, and D. Vlachos.
    Spatially adaptive lattice coarse-grained Monte Carlo simulations for diffusion of interacting molecules.
    J. Chem. Phys., 121(22):11420-11431, 2004.
  • [8] A. Chatterjee, M. Katsoulakis, and D. Vlachos.
    Spatially adaptive grand canonical ensemble Monte Carlo simulations.
    Phys. Rev. E, 71, 2005.
  • [9] A Chatterjee and DG Vlachos.
    Multiscale spatial Monte Carlo simulations: Multigriding, computational singular perturbation, and hierarchical stochastic closures.
    J. Chem. Phys., 124(6), FEB 14 2006.
  • [10] A. Chatterjee and D.G. Vlachos.
    An overview of spatial microscopic and accelerated kinetic monte carlo methods.
    J. Comput-Aided Mater. Des., 14(2):253-308, 2007.
  • [11] Jianguo Dai, W. D. Seider, and T. Sinno.
    Coarse-grained lattice kinetic Monte Carlo simulation of systems of strongly interacting particles.
    J. Chem. Phys., 128(19):194705, 2008.
  • [12] R. L. Dobrushin and S. B. Shlosman, Completely analytical interactions: constructive description, J. Statist. Phys. 46 (1987), no. 5-6, 983–1014. MR 893129, 10.1007/BF01011153
  • [13] E. Espanol, M. Serrano, and Zuniga.
    Coarse-graining of a fluid and its relation with dissipasive particle dynamics and smoothed particle dynamics.
    Int. J. Modern Phys. C, 8(4):899-908, 1997.
  • [14] P. Espanol and P. Warren.
    Statistics-mechanics of dissipative particle dynamics.
    Europhys. Lett., 30(4):191-196, 1995.
  • [15] Francesca Fierro and Andreas Veeser, On the a posteriori error analysis for equations of prescribed mean curvature, Math. Comp. 72 (2003), no. 244, 1611–1634 (electronic). MR 1986796, 10.1090/S0025-5718-03-01507-2
  • [16] H. Fukunaga, J. Takimoto, and M. Doi.
    A coarse-graining procedure for flexible polymer chains with bonded and nonbonded interactions.
    J. Chem. Phys., 116(18):8183-8190, 2002.
  • [17] N. Goldenfeld.
    Lectures on Phase Transitions and the Renormalization Group, volume 85.
    Addison-Wesley, New York, 1992.
  • [18] G. Hadjipanayis, editor.
    Magnetic Hysteresis in Novel Magnetic Materials, volume 338 of NATO ASI Series E, Dordrecht, The Netherlands, 1997. Kluwer Academic Publishers.
  • [19] V.A. Harmandaris, N.P. Adhikari, N.F.A. van der Vegt, and K. Kremer.
    Hierarchical modeling of polystyrene: From atomistic to coarse-grained simulations.
    Macromolecules, 39:6708-6719, 2006.
  • [20] L. Kadanoff.
    Scaling laws for Ising models near $ t_c$.
    Physics, 2:263, 1966.
  • [21] Markos A. Katsoulakis, Andrew J. Majda, and Dionisios G. Vlachos, Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems, J. Comput. Phys. 186 (2003), no. 1, 250–278. MR 1967368, 10.1016/S0021-9991(03)00051-2
  • [22] M. A. Katsoulakis, L. Rey-Bellet, P. Plecháč, and D. K.Tsagkarogiannis.
    Mathematical strategies in the coarse-graining of extensive systems: error quantification and adaptivity.
    J. Non Newt. Fluid Mech., 152:101-112, 2008.
  • [23] Markos A. Katsoulakis, Petr Plecháč, and Luc Rey-Bellet, Numerical and statistical methods for the coarse-graining of many-particle stochastic systems, J. Sci. Comput. 37 (2008), no. 1, 43–71. MR 2442973, 10.1007/s10915-008-9216-6
  • [24] Markos A. Katsoulakis, Petr Plecháč, Luc Rey-Bellet, and Dimitrios K. Tsagkarogiannis, Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems, M2AN Math. Model. Numer. Anal. 41 (2007), no. 3, 627–660. MR 2355714, 10.1051/m2an:2007032
  • [25] Markos A. Katsoulakis, Petr Plecháč, and Alexandros Sopasakis, Error analysis of coarse-graining for stochastic lattice dynamics, SIAM J. Numer. Anal. 44 (2006), no. 6, 2270–2296 (electronic). MR 2272594, 10.1137/050637339
  • [26] Markos A. Katsoulakis and José Trashorras, Information loss in coarse-graining of stochastic particle dynamics, J. Stat. Phys. 122 (2006), no. 1, 115–135. MR 2203785, 10.1007/s10955-005-8063-1
  • [27] K. Kremer and F. Muller-Plathe.
    Multiscale problems in polymer science: Simulation approaches.
    MRS Bulletin, 26(3):205-210, 2001.
  • [28] Omar Lakkis and Ricardo H. Nochetto, A posteriori error analysis for the mean curvature flow of graphs, SIAM J. Numer. Anal. 42 (2005), no. 5, 1875–1898 (electronic). MR 2139228, 10.1137/S0036142903430207
  • [29] David P. Landau and Kurt Binder, A guide to Monte Carlo simulations in statistical physics, Cambridge University Press, Cambridge, 2000. MR 1781083
  • [30] A. P. Lyubartsev, M. Karttunen, P. Vattulainen, and A. Laaksonen.
    On coarse-graining by the inverse monte carlo method: Dissipative particle dynamics simulations made to a precise tool in soft matter modeling.
    Soft Materials, 1(1):121-137, 2003.
  • [31] F. Müller-Plathe.
    Coarse-graining in polymer simulation: from the atomistic to the mesoscale and back.
    Chem. Phys. Chem., 3:754, 2002.
  • [32] Enzo Olivieri, On a cluster expansion for lattice spin systems: a finite-size condition for the convergence, J. Statist. Phys. 50 (1988), no. 5-6, 1179–1200. MR 951074, 10.1007/BF01019160
  • [33] Enzo Olivieri and Pierre Picco, Cluster expansion for 𝑑-dimensional lattice systems and finite-volume factorization properties, J. Statist. Phys. 59 (1990), no. 1-2, 221–256. MR 1049968, 10.1007/BF01015569
  • [34] I. Pivkin and G. Karniadakis.
    Coarse-graining limits in open and wall-bounded dissipative particle dynamics systems.
    J. Chem. Phys., 124:184101, 2006.
  • [35] R. Plass, J.A. Last, N.C. Bartelt, and G.L. Kellogg.
    Self-assembled domain patterns.
    Nature, 412:875, 2001.
  • [36] M. Praprotnik, S. Matysiak, L. Delle Site, K. Kremer, and C. Clementi.
    Adaptive resolution simulation of liquid water.
    J. Physics: Condensed Matter, 19(29):292201 (10pp), 2007.
  • [37] M. Seul and D. Andelman.
    Domain shapes and patterns: the phenomenology of modulated phases.
    Science, 267:476-483, 1995.
  • [38] Barry Simon, The statistical mechanics of lattice gases. Vol. I, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993. MR 1239893
  • [39] José Trashorras and Dimitrios K. Tsagkarogiannis, From mesoscale back to microscale: reconstruction schemes for coarse-grained stochastic lattice systems, SIAM J. Numer. Anal. 48 (2010), no. 5, 1647–1677. MR 2733093, 10.1137/080722382
  • [40] W. Tschöp, K. Kremer, O. Hahn, J. Batoulis, and T. Bürger.
    Simulation of polymer melts. II. From coarse-grained models back to atomistic description.
    Acta Polym., 49:75, 1998.
  • [41] G.A. Voth.
    Coarse-Graining of Condensed Phase and Biomolecular Systems.
    CRC Press, Boca Raton, FL, 2009.

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Additional Information

Markos A. Katsoulakis
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003 and Department of Applied Mathematics, University of Crete and Foundation of Research – and — Technology-Hellas, Greece
Email: markos@math.umass.edu

Petr Plecháč
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email: plechac@math.udel.edu

Luc Rey-Bellet
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
Email: lr7q@math.umass.edu

Dimitrios K. Tsagkarogiannis
Affiliation: Hausdorff Center for Mathematics, University of Bonn, D-53115 Bonn, Germany
Email: dtsagkaro@gmail.com

DOI: https://doi.org/10.1090/S0025-5718-2014-02806-8
Keywords: Coarse-graining, lattice spin systems, Monte Carlo method, Gibbs measure, cluster expansion, renormalization group map, sub-grid scale modeling, multi-body interactions
Received by editor(s): March 8, 2010
Received by editor(s) in revised form: March 22, 2011, and November 9, 2011
Published electronically: March 25, 2014
Additional Notes: The research of the first author was supported by the National Science Foundation through grants NSF-DMS-0715125, the CDI -Type II award NSF-CMMI-0835673, and the European Commission FP7-REGPOT-2009-1 project “Archimedes Center for Modeling, Analysis and Computation”
The research of the second author was partially supported by the National Science Foundation under grant NSF-DMS-0813893 and by the Office of Advanced Scientific Computing Research, U.S. Department of Energy under DE-SC0001340; the work was partly done at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725
The research of the third author was partially supported by grant NSF-DMS-06058
The research of the fourth author has been supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Program
Article copyright: © Copyright 2014 American Mathematical Society