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Coarse-graining schemes and a posteriori error estimatesfor stochastic lattice systems

Published online by Cambridge University Press:  02 August 2007

Markos A. Katsoulakis ,
Petr Plecháč ,
Luc Rey-Bellet  and
Dimitrios K. Tsagkarogiannis
Markos A. Katsoulakis
Affiliation:
Department of Mathematics, University of Massachusetts, USA. markos@math.umass.edu; lr7q@math.umass.edu
Petr Plecháč
Affiliation:
Department of Mathematics, University of Tennessee, USA. plechac@math.utk.edu
Luc Rey-Bellet
Affiliation:
Department of Mathematics, University of Massachusetts, USA. markos@math.umass.edu; lr7q@math.umass.edu
Dimitrios K. Tsagkarogiannis
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Germany. tsagkaro@mis.mpg.de
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Abstract

The primary objective of this work is to develop coarse-grainingschemes for stochastic many-body microscopic models and quantify theireffectiveness in terms of a priori and a posteriori error analysis. Inthis paper we focus on stochastic lattice systems ofinteracting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grainedapproximation 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 forthe coarse-grained approximations that can be used for the construction ofadaptive coarse-graining methods.
We present a number of numerical examples that demonstrate that thecoarse-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 improvepredictions of earlier coarse-graining schemes for short-range interactions.

Keywords

Coarse-graininga posteriori error estimaterelative entropylattice spin systemsMonte Carlo methodGibbs measurecluster expansionrenormalization group map.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 3 , May 2007 , pp. 627 - 660
Copyright
© EDP Sciences, SMAI, 2007

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References

Bricmont, J., Kupiainen, A. and Lefevere, R., Renormalization group pathologies and the definition of Gibbs states. Comm. Math. Phys. 194 (1998) 359–388. CrossRef
Cammarota, C., Decay of correlations for infinite range interactions in unbounded spin systems. Comm. Math. Phys. 85 (1982) 517–528. CrossRef
Chatterjee, A., Katsoulakis, M. and Vlachos, D., Spatially adaptive lattice coarse-grained Monte Carlo simulations for diffusion of interacting molecules. J. Chem. Phys. 121 (2004) 11420–11431. CrossRef
Chatterjee, A., Katsoulakis, M. and Vlachos, D., Spatially adaptive grand canonical ensemble Monte Carlo simulations. Phys. Rev. E 71 (2005) 026702.
T.M. Cover and J.A. Thomas, Elements of Information Theory. John Wiley and Sons, Inc. (1991).
Gallavotti, G.A. and Miracle-Sole, S., Correlation functions of a lattice system. Comm. Math. Phys. 7 (1968) 274–288. CrossRef
N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Volume 85. Addison-Wesley, New York (1992).
Gruber, C. and Kunz, H., General properties of polymer systems. Comm. Math. Phys. 22 (1971) 133–161. CrossRef
Hildebrand, M. and Mikhailov, A.S., Mesoscopic modeling in the kinetic theory of adsorbates. J. Chem. Phys. 100 (1996) 19089. CrossRef
Ismail, A.E., Rutledge, G.C. and Stephanopoulos, G., Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamics properties. J. Chem. Phys. 118 (2003) 4414–4424. CrossRef
Ismail, A.E., Rutledge, G.C. and Stephanopoulos, G., Multiresolution analysis in statistical mechanics. II. Wavelet transform as a basis for Monte Carlo simulations on lattices. J. Chem. Phys. 118 (2003) 4424. CrossRef
Kadanoff, L., Scaling laws for Ising models near t c . Physics 2 (1966) 263.
Katsoulakis, M. and Trashorras, J., Information loss in coarse-graining of stochastic particle dynamics. J. Statist. Phys. 122 (2006) 115–135. CrossRef
Katsoulakis, M., Majda, A. and Vlachos, D., Coarse-grained stochastic processes for microscopic lattice systems. Proc. Natl. Acad. Sci. 100 (2003) 782–782. CrossRef
Katsoulakis, M.A., Majda, A.J. and Vlachos, D.G., Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems. J. Comp. Phys. 186 (2003) 250–278. CrossRef
M.A. Katsoulakis, P. Plecháč, L. Rey-Bellet and D.K. Tsagkarogiannis, Coarse-graining schemes for lattice systems with short and long range interactions. (In preparation).
Katsoulakis, M.A., Plecháč, P. and Sopasakis, A., Error analysis of coarse-graining for stochastic lattice dynamics. SIAM J. Numer. Anal. 44 (2006) 2270. CrossRef
D.A. Lavis and G.M. Bell, Statistical Mechanics of Lattice Systems, Volume I. Springer Verlag (1999).
Mayer, J.E., Integral equations between distribution functions of molecules. J. Chem. Phys. 15 (1947) 187–201. CrossRef
Peierls, R., Ising's, On model of ferromagnetism. Proc. Camb. Philos. Soc. 32 (1936) 477–481. CrossRef
Pivkin, I.V. and Karniadakis, G.E., Coarse-graining limits in open and wall-bounded dissipative particle dynamics systems. J. Chem. Phys. 124 (2006) 184101. CrossRef
Procacci, A., De Lima, B.N.B. and Scoppola, B., A remark on high temperature polymer expansion for lattice systems with infinite range pair interactions. Lett. Math. Phys. 45 (1998) 303–322. CrossRef
B. Simon, The Statistical Mechanics of Lattice Gases, Vol. I. Princeton series in Physics (1993).
Szepessy, A., Tempone, R. and Zouraris, G.E., Adaptive weak approximation of stochastic differential equations. Comm. Pure Appl. Math. 54 (2001) 1169–1214. CrossRef
van Enter, A.C.D., Fernández, R. and Sokal, A.D., Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Statist. Phys. 72 (1993) 879–1167. CrossRef