Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems
Katsoulakis, Markos A. ; Plecháč, Petr ; Rey-Bellet, Luc ; Tsagkarogiannis, Dimitrios K.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007), p. 627-660 / Harvested from Numdam

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. 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.

Publié le : 2007-01-01
DOI : https://doi.org/10.1051/m2an:2007032
Classification:  65C05,  65C20,  82B20,  82B80,  82-08
@article{M2AN_2007__41_3_627_0,
     author = {Katsoulakis, Markos A. and Plech\'a\v c, Petr and Rey-Bellet, Luc and Tsagkarogiannis, Dimitrios K.},
     title = {Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {41},
     year = {2007},
     pages = {627-660},
     doi = {10.1051/m2an:2007032},
     mrnumber = {2355714},
     zbl = {pre05289387},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2007__41_3_627_0}
}
Katsoulakis, Markos A.; Plecháč, Petr; Rey-Bellet, Luc; Tsagkarogiannis, Dimitrios K. Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) pp. 627-660. doi : 10.1051/m2an:2007032. http://gdmltest.u-ga.fr/item/M2AN_2007__41_3_627_0/

[1] J. Bricmont, A. Kupiainen and R. Lefevere, Renormalization group pathologies and the definition of Gibbs states. Comm. Math. Phys. 194 (1998) 359-388. | Zbl 0914.60089

[2] C. Cammarota, Decay of correlations for infinite range interactions in unbounded spin systems. Comm. Math. Phys. 85 (1982) 517-528.

[3] A. Chatterjee, M. Katsoulakis and D. Vlachos, Spatially adaptive lattice coarse-grained Monte Carlo simulations for diffusion of interacting molecules. J. Chem. Phys. 121 (2004) 11420-11431.

[4] A. Chatterjee, M. Katsoulakis and D. Vlachos, Spatially adaptive grand canonical ensemble Monte Carlo simulations. Phys. Rev. E 71 (2005) 026702.

[5] T.M. Cover and J.A. Thomas, Elements of Information Theory. John Wiley and Sons, Inc. (1991). | MR 1122806 | Zbl 0762.94001

[6] G.A. Gallavotti and S. Miracle-Sole, Correlation functions of a lattice system. Comm. Math. Phys. 7 (1968) 274-288.

[7] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Volume 85. Addison-Wesley, New York (1992).

[8] C. Gruber and H. Kunz, General properties of polymer systems. Comm. Math. Phys. 22 (1971) 133-161.

[9] M. Hildebrand and A.S. Mikhailov, Mesoscopic modeling in the kinetic theory of adsorbates. J. Chem. Phys. 100 (1996) 19089.

[10] A.E. Ismail, G.C. Rutledge and G. Stephanopoulos, Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamics properties. J. Chem. Phys. 118 (2003) 4414-4424.

[11] A.E. Ismail, G.C. Rutledge and G. Stephanopoulos, Multiresolution analysis in statistical mechanics. II. Wavelet transform as a basis for Monte Carlo simulations on lattices. J. Chem. Phys. 118 (2003) 4424.

[12] L. Kadanoff, Scaling laws for Ising models near t c . Physics 2 (1966) 263.

[13] M. Katsoulakis and J. Trashorras, Information loss in coarse-graining of stochastic particle dynamics. J. Statist. Phys. 122 (2006) 115-135. | Zbl 1127.82043

[14] M. Katsoulakis, A. Majda and D. Vlachos, Coarse-grained stochastic processes for microscopic lattice systems. Proc. Natl. Acad. Sci. 100 (2003) 782-782. | Zbl 1063.82033

[15] M.A. Katsoulakis, A.J. Majda and D.G. Vlachos, Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems. J. Comp. Phys. 186 (2003) 250-278. | Zbl 1034.82053

[16] 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).

[17] M.A. Katsoulakis, P. Plecháč and A. Sopasakis, Error analysis of coarse-graining for stochastic lattice dynamics. SIAM J. Numer. Anal. 44 (2006) 2270. | MR 2272594 | Zbl 1140.82035

[18] D.A. Lavis and G.M. Bell, Statistical Mechanics of Lattice Systems, Volume I. Springer Verlag (1999). | Zbl 0925.82001

[19] J.E. Mayer, Integral equations between distribution functions of molecules. J. Chem. Phys. 15 (1947) 187-201.

[20] R. Peierls, On Ising's model of ferromagnetism. Proc. Camb. Philos. Soc. 32 (1936) 477-481. | Zbl 0014.33604

[21] I.V. Pivkin and G.E. Karniadakis, Coarse-graining limits in open and wall-bounded dissipative particle dynamics systems. J. Chem. Phys. 124 (2006) 184101.

[22] A. Procacci, B.N.B. De Lima and B. Scoppola, A remark on high temperature polymer expansion for lattice systems with infinite range pair interactions. Lett. Math. Phys. 45 (1998) 303-322. | Zbl 0929.60088

[23] B. Simon, The Statistical Mechanics of Lattice Gases, Vol. I. Princeton series in Physics (1993). | MR 1239893 | Zbl 0804.60093

[24] A. Szepessy, R. Tempone and G.E. Zouraris, Adaptive weak approximation of stochastic differential equations. Comm. Pure Appl. Math. 54 (2001) 1169-1214. | Zbl 1024.60028

[25] A.C.D. Van Enter, R. Fernández and A.D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Statist. Phys. 72 (1993) 879-1167. | Zbl 1101.82314