Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Coarse-graining schemes for stochastic lattice systems with short and long-range interactions
HTML articles powered by AMS MathViewer

by Markos A. Katsoulakis, Petr Plecháč, Luc Rey-Bellet and Dimitrios K. Tsagkarogiannis PDF
Math. Comp. 83 (2014), 1757-1793 Request permission

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.
References
  • 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.
  • 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, DOI 10.1137/080713276
  • 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
  • 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, DOI 10.1023/A:1004620929047
  • 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, DOI 10.1023/A:1018631710626
  • 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
  • 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.
  • A. Chatterjee, M. Katsoulakis, and D. Vlachos. Spatially adaptive grand canonical ensemble Monte Carlo simulations. Phys. Rev. E, 71, 2005.
  • 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.
  • 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.
  • 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.
  • R. L. Dobrushin and S. B. Shlosman, Completely analytical interactions: constructive description, J. Statist. Phys. 46 (1987), no. 5-6, 983–1014. MR 893129, DOI 10.1007/BF01011153
  • 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.
  • P. Espanol and P. Warren. Statistics-mechanics of dissipative particle dynamics. Europhys. Lett., 30(4):191–196, 1995.
  • 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. MR 1986796, DOI 10.1090/S0025-5718-03-01507-2
  • 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.
  • N. Goldenfeld. Lectures on Phase Transitions and the Renormalization Group, volume 85. Addison-Wesley, New York, 1992.
  • G. Hadjipanayis, editor. Magnetic Hysteresis in Novel Magnetic Materials, volume 338 of NATO ASI Series E, Dordrecht, The Netherlands, 1997. Kluwer Academic Publishers.
  • 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.
  • L. Kadanoff. Scaling laws for Ising models near $t_c$. Physics, 2:263, 1966.
  • 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, DOI 10.1016/S0021-9991(03)00051-2
  • 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.
  • 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, DOI 10.1007/s10915-008-9216-6
  • 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, DOI 10.1051/m2an:2007032
  • 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. MR 2272594, DOI 10.1137/050637339
  • 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, DOI 10.1007/s10955-005-8063-1
  • K. Kremer and F. Muller-Plathe. Multiscale problems in polymer science: Simulation approaches. MRS Bulletin, 26(3):205–210, 2001.
  • 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. MR 2139228, DOI 10.1137/S0036142903430207
  • David P. Landau and Kurt Binder, A guide to Monte Carlo simulations in statistical physics, Cambridge University Press, Cambridge, 2000. MR 1781083
  • 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.
  • F. Müller-Plathe. Coarse-graining in polymer simulation: from the atomistic to the mesoscale and back. Chem. Phys. Chem., 3:754, 2002.
  • 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, DOI 10.1007/BF01019160
  • 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, DOI 10.1007/BF01015569
  • I. Pivkin and G. Karniadakis. Coarse-graining limits in open and wall-bounded dissipative particle dynamics systems. J. Chem. Phys., 124:184101, 2006.
  • R. Plass, J.A. Last, N.C. Bartelt, and G.L. Kellogg. Self-assembled domain patterns. Nature, 412:875, 2001.
  • 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.
  • M. Seul and D. Andelman. Domain shapes and patterns: the phenomenology of modulated phases. Science, 267:476–483, 1995.
  • Barry Simon, The statistical mechanics of lattice gases. Vol. I, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993. MR 1239893, DOI 10.1515/9781400863433
  • 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, DOI 10.1137/080722382
  • 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.
  • G.A. Voth. Coarse-Graining of Condensed Phase and Biomolecular Systems. CRC Press, Boca Raton, FL, 2009.
Similar Articles
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
  • 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
  • © Copyright 2014 American Mathematical Society
  • Journal: Math. Comp. 83 (2014), 1757-1793
  • MSC (2010): Primary 65C05, 65C20, 82B20, 82B80, 82-08
  • DOI: https://doi.org/10.1090/S0025-5718-2014-02806-8
  • MathSciNet review: 3194129