Journal of Statistical Physics

, Volume 119, Issue 1–2, pp 347–389 | Cite as

Mesoscopic Modeling for Continuous Spin Lattice Systems: Model Problems and Micromagnetics Applications

  • Markos A. Katsoulakis
  • Petr Plecháč
  • Dimitrios K. Tsagkarogiannis
Article

Abstract

In this paper we derive deterministic mesoscopic theories for model continuous spin lattice systems both at equilibrium and non-equilibrium in the presence of thermal fluctuations. The full magnetic Hamiltonian that includes singular integral (dipolar) interactions is also considered at equilibrium. The non-equilibrium microscopic models we consider are relaxation-type dynamics arising in kinetic Monte Carlo or Langevin-type simulations of lattice systems. In this context we also employ the derived mesoscopic models to study the relaxation of such algorithms to equilibrium

Keywords

Heisenberg spin lattice system Kac potential large deviations statistical equilibrium Monte Carlo methods relaxation dynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Landau D., Binder. K. (2000). A Guide to Monte Carlo Simulations in Statistical Physics Cambridge University PressGoogle Scholar
  2. O’Handley R.C. (2000). Modern Magnetic Materials: Principles and Applications (Wiley)Google Scholar
  3. Hubert A., Schafer R. (1998). Magnetic domains (Springer, 1998)Google Scholar
  4. Boucher, C., Ellis, R.S., Turkington, B. 2000Derivation of maximum entropy principles in two-dimensional turbulence via large deviationsJ. Stat. Phy.9812351278CrossRefGoogle Scholar
  5. Trashorras, J. 2003Principes de grandes déviations pour des processus de “coarse graining”C. R. Acad. Sci. Paris – Mathematique Ser. I3366974Google Scholar
  6. Lebowitz, J.L., Orlandi, E., Presutti, E. 1991A particle model for spinodal decompositionJ. Stat. Phys.63933974CrossRefGoogle Scholar
  7. Masi, A., Orlandi, E., Presutti, E., Triolo, L. 1994Glauber evolution with the Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamicsNonlinearity.7633696CrossRefGoogle Scholar
  8. Katsoulakis, M.A., Souganidis, P.E. 1997Stochastic Ising models and anisotropic front propagationJ. Stat. Phys.876389Google Scholar
  9. Giacomin G., Lebowitz J. L., Presutti E. Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems in Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys Monogr, 64, (Am Math. Soc., Providence, RI, 1999). pp. 107–152Google Scholar
  10. Lanford O. Entropy and equilibrium states in classical statistical mechanics, Statistical Mechanics and Mathematical problems, volume 20 Lecture Notes in Physics, in A. Lenard ed. (Springer-Verlag, 1971), pp. 1–113Google Scholar
  11. Lewis, J.T., Pfister, C.E., Sullivan., W.G. 1994The equivalence of ensembles for lattice systems: some examples and a counterexampleJ. Stat. Phys.77397419Google Scholar
  12. Ellis, R.S., Heaven, K., Turkington.,  2000Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensemblesJ. Stat. Phys.10199991064CrossRefGoogle Scholar
  13. Dupuis, P., Ellis, R.S. 1997A weak convergence approach to the theory of large deviationsJohn Wiley & Sons Inc.New YorkA Wiley-Interscience PublicationGoogle Scholar
  14. Eisele, T., Ellis, R.S. 1983Symmetry breaking and random waves for magnetic systems on a circle ZWahrsch. Verw. Gebiete.63297348CrossRefGoogle Scholar
  15. Cassandro, M., Orlandi, E., Presutti, E. 1993Interfaces and typical Gibbs configurations for one-dimensional Kac potentialsProbab. Theory Relat. Fields.965796CrossRefGoogle Scholar
  16. Bovier, A., Zahradník, M. 1997The low-temperature phase of Kac-Ising modelsJ. Stat. Phys.87311332Google Scholar
  17. Cassandro, M., Orlandi, E., Picco, P. 1999Typical configurations for one-dimensional random field Kac model AnnProbab.2714141467CrossRefGoogle Scholar
  18. Boucher, C., Ellis, R.S., Turkington, B. 1999Spatializing random measures: Doubly indexed processes and the large deviation principleAnn. Probab.27297324CrossRefGoogle Scholar
  19. Stein E.M. (1970). Singular Integrals and Differentiability Properties of Functions PrincetonGoogle Scholar
  20. James, R.D., Müller, S. 1994Internal variables and fine-scale oscillations in micromagneticsContin. Mech. Thermodyn.6291336CrossRefGoogle Scholar
  21. Katsoulakis M., Plechá č P., Trashorras J., Tsagkarogiannis D., Large deviations for lattice systems with singular interactions, in preparationGoogle Scholar
  22. Jordan, R., Kinderlehrer, D., Otto, F. 1998The variational formulation of the Fokker– Planck equation SIAM JMath. Anal.29117electronicCrossRefGoogle Scholar
  23. Landau L.D., Lifschitz E.M. (1935). On the theory of dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8Google Scholar
  24. Kittel, C. 1949Physical theory of ferromagnetic domainsRev. Mod. Phys.21541583CrossRefGoogle Scholar
  25. Brown, W.F. 1963MicromagneticsWileyNew YorkGoogle Scholar
  26. Brenier, Y. 1989Un algorithme rapide pour le calcul de transformees de Legendre–Fenchel discretes CR. Acad. Sci. Paris Serie I.308587589Google Scholar
  27. De Masi A., Presutti E. (1991). Mathematical Methods for Hydrodynamic Limits, volume 1501 of Lecture Notes in Mathematics Springer-Verlag,Google Scholar
  28. Rabinowitz, P.H. 1989Periodic and heteroclinic orbits for a periodic Hamiltonian systemAnn. Inst. H. Poincaré Anal. Non Linéaire.6331346Google Scholar
  29. Auerbach, S.M. 2000Theory and simulation of jump dynamics diffusion and phase equilibrium in nanoporesInt. Rev. Phys. Chem.19155CrossRefGoogle Scholar
  30. Katsoulakis M.A., Majda A.J., Vlachos D. Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems J. Comp. Phys. 186:250–278Google Scholar
  31. Gidas B. (1995). Metropolis-type Monte Carlo simulation algorithms and simulated annealing, in Topics in Contemporary Probability and its Applications, J. Laurie Snell, ed. CRC PressGoogle Scholar
  32. Giacomin, G., Lebowitz, J.L. 1997Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limitsJ. Stat. Phys.873761Google Scholar
  33. Kipnis C., Landim C. (1999). Scaling Limits of Interacting Particle Systems (Springer-Verlag,)Google Scholar
  34. Shiryaev A.N. (1996). Probability (Springer, Graduate Textes in Mathematics, No 95,)Google Scholar
  35. Cercignani C. (1988).. The Boltzmann Equation and Its Applications, volume 67 Applied Mathematical Sciences (Springer-Verlag,)Google Scholar
  36. Comets, F., Eisele, Th. 1988Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interacting modelCommun. Math. Phys.118531567CrossRefGoogle Scholar
  37. Kato T. (1984). Perturbation Theory for Linear Operators (Springer-Verlag)Google Scholar
  38. Arnold, A., Markowich, P., Toscani, G., Unterreiter, A. 2001On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equationsCommun. Partial Diff. Equations.2643100CrossRefGoogle Scholar
  39. Stroock, D.W., Varadhan, S.R.S. 1979Multidimensional Diffusion ProcessesSpringer-VerlagNew YorkGoogle Scholar
  40. Cépa, E., Lépingle, D. 1997Diffusing particles with electrostatic repulsionProb. Theory Relat. Fields.107429449CrossRefGoogle Scholar
  41. Katsoulakis, M., Majda, A., Vlachos, D. 2003Coarse-grained stochastic processes for microscopic lattice systemsProc Natl. Acad. Sci.100782CrossRefPubMedGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Markos A. Katsoulakis
    • 1
  • Petr Plecháč
    • 2
  • Dimitrios K. Tsagkarogiannis
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations