Skip to main content
SpringerLink
Log in

On the Convergence of Kikuchi’s Natural Iteration Method

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this article we investigate on the convergence of the natural iteration method, a numerical procedure widely employed in the statistical mechanics of lattice systems, to minimize Kikuchi’s cluster variational free energies. We discuss a sufficient condition for the convergence, based on the coefficients of the cluster entropy expansion, depending on the lattice geometry. We also show that such a condition is satisfied for many lattices usually studied in applications. Finally, we consider a recently proposed class of methods for the minimization of Kikuchi functionals, showing that the natural iteration method turns out as a particular instance of that class.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Kikuchi (1951) Phys. Rev 81 988 Occurrence Handle10.1103/PhysRev.81.988

    Article  Google Scholar 

  2. T. Morita (1972) J. Math. Phys 13 115 Occurrence Handle10.1063/1.1665840

    Article  Google Scholar 

  3. A.G. Schlijper (1983) Phys. Rev. B 27 6841 Occurrence Handle10.1103/PhysRevB.27.6841

    Article  Google Scholar 

  4. G. An (1988) J. Stat. Phys 52 727 Occurrence Handle10.1007/BF01019726

    Article  Google Scholar 

  5. R. Kikuchi (1994) Progr. Theor. Phys. Suppl 115 1

    Google Scholar 

  6. C. Giaconia V. Pagot R. Tétot (2000) Physica C 330 44 Occurrence Handle1:CAS:528:DC%2BD3cXht1equrY%3D

    CAS  Google Scholar 

  7. M. Asta J.J. Hoyt (2000) Acta Mater 48 1089 Occurrence Handle1:CAS:528:DC%2BD3cXhsFagsbk%3D

    CAS  Google Scholar 

  8. C.G. Schon G. Inden (1996) Scripta Mater 34 1035 Occurrence Handle10.1016/1359-6462(95)00623-0

    Article  Google Scholar 

  9. E. Kentzinger et al. (2000) Phys. Rev. B 61 14975 Occurrence Handle1:CAS:528:DC%2BD3cXjslWnsrY%3D

    CAS  Google Scholar 

  10. E. Lopez-Sandoval J.L. Moran-Lopez F. Aguilera-Granja (1999) Solid State Comm 112 437 Occurrence Handle1:CAS:528:DyaK1MXmvFegt7Y%3D

    CAS  Google Scholar 

  11. W.A. Oates F. Zhang S.L. Chen Y.A. Chang (1999) Phys. Rev. B 59 11221 Occurrence Handle1:CAS:528:DyaK1MXis1Shuro%3D

    CAS  Google Scholar 

  12. C. Buzano M. Pretti (1997) Phys. Rev. B 56 636 Occurrence Handle1:CAS:528:DyaK2sXksFOhtLw%3D

    CAS  Google Scholar 

  13. S. Lapinskas E.E. Tornau V.E. Zubkus A. Rosengren (1998) Surf. Sci 414 44 Occurrence Handle1:CAS:528:DyaK1cXmsVOiu7k%3D

    CAS  Google Scholar 

  14. E. Clouet M. Nastar C. Sigli (2004) Phys. Rev. B 69 64109 Occurrence Handle10.1103/PhysRevB.69.064109

    Article  Google Scholar 

  15. C.G. Schon G. Inden (2001) Comput. Mat. Sci 20 98 Occurrence Handle1:CAS:528:DC%2BD3MXntFOjsw%3D%3D

    CAS  Google Scholar 

  16. A. Pelizzola (2000) Phys. Rev. E 61 4915 Occurrence Handle1:CAS:528:DC%2BD3cXivFCktL0%3D

    CAS  Google Scholar 

  17. A. Pelizzola (1994) Phys. Rev. E 49 R2503 Occurrence Handle10.1103/PhysRevE.49.R2503

    Article  Google Scholar 

  18. M. Katori M. Suzuki (1994) Progr. Theor. Phys. Suppl 115 83

    Google Scholar 

  19. M. Katori M. Suzuki (1988) J. Phys. Soc 57 3753

    Google Scholar 

  20. K. Tanaka J. Inoue D.M. Titterington (2003) J. Phys. A 36 11023 Occurrence HandleMR2025242

    MathSciNet  Google Scholar 

  21. F.R. Kschnischang B.J. Frey H.-A. Loeliger (2002) IEEE. Trans. Inform. Theory 47 498 Occurrence Handle10.1109/18.910572

    Article  Google Scholar 

  22. B.J. Frey (1998) Graphical Models for Machine Learning and Digital Communication MIT press MA

    Google Scholar 

  23. J.S. Yedidia, W.T. Freeman, and Y.Weiss, Mitsubishi Electric R.L. Technical Report 35 (2002).

  24. A.L. Yuille (2001) Neural Comp 14 1691 Occurrence Handle10.1162/08997660260028674

    Article  Google Scholar 

  25. T. Heskes, K. Albers, and B. Kappen, UAI-2003 proceedings, pp. 313–320 (2003).

  26. M. Pretti A. Pelizzola (2003) J. Phys. A 36 11201

    Google Scholar 

  27. R. Kikuchi (1974) J. Chem. Phys 60 1071 Occurrence Handle1:CAS:528:DyaE2cXlsVWjtr8%3D

    CAS  Google Scholar 

  28. M. Pretti (2003) J. Stat. Phys 111 993 Occurrence Handle10.1023/A:1022862618478

    Article  Google Scholar 

  29. T.C. King H.H. Chen (1999) Physica. A 267 153 Occurrence Handle1:CAS:528:DyaK1MXivFelurk%3D

    CAS  Google Scholar 

  30. R. Kikuchi S.G. Brush (1967) J. Chem. Phys 47 195 Occurrence Handle1:CAS:528:DyaF2sXksFOgtr0%3D

    CAS  Google Scholar 

  31. A. Pelizzola M. Pretti (1999) Phys. Rev. B 60 10134 Occurrence Handle1:CAS:528:DyaK1MXmsVelt7s%3D

    CAS  Google Scholar 

  32. R. Kikuchi (1976) J. Chem. Phys 65 4545 Occurrence Handle1:CAS:528:DyaE2sXnvV2nuw%3D%3D

    CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto Nazionale per la Fisica della Materia (INFM) and Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129, Torino, Italy

    Marco Pretti

Authors
  1. Marco Pretti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Pretti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pretti, M. On the Convergence of Kikuchi’s Natural Iteration Method. J Stat Phys 119, 659–675 (2005). https://doi.org/10.1007/s10955-005-4426-x

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-005-4426-x

Keywords

Search

Navigation