Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Analytical approximations for the hierarchically constrained kinetic Ising chain

  • 53 Accesses

  • 33 Citations


The hierarchically constrained kinetic Ising model in one dimension is reviewed, and the results of several analytical approaches to the model are presented. Two standard approximation schemes, an effective-medium approximation and a mode-coupling approximation, are shown to fail. A new class of approximations, termed cluster approximations, is better suited for the model. It yields good results for the spin autocorrelation function, and also elucidates important general properties of the model—its connection with defect-diffusion models and the asymptotic long-time behavior of the autocorrelation function.

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


  1. 1.

    J. Jäckle and S. Eisinger, A hierarchically constrained kinetic Ising model,Z. Phys. B-Condensed Matter 84:115–124 (1991).

  2. 2.

    E. Abrahams, R. G. Palmer, D. L. Stein, and P. W. Anderson, Models of hierarchically constrained dynamics for glassy relaxation,Phys. Rev. Lett. 53(10):958–961 (1984).

  3. 3.

    R. J. Glauber, Time-dependent statistics of the Ising model,J. Math. Phys. 4(2):294–307 (1963).

  4. 4.

    C. A. Condat, Solution to the Glarum model with a finite relaxation rate,Z. Phys. B-Condensed Matter 77:313–320 (1989).

  5. 5.

    P. Borderwijk, Defect-diffusion models of dielectric relaxation,Chem. Phys. Lett. 32(3):592–596 (1975).

  6. 6.

    S. Eisinger, A hierarchical spin model, Ph.D. thesis, University of Konstanz (1992).

  7. 7.

    J. P. Hansen and I. R. McDonald,Theory of Simple Liquids, 2nd. ed. (Academic Press, 1986).

  8. 8.

    J. Jäckle and D. Sappelt, Test of analytical approximations for kinetic Ising models with sharp blocking transitions,Physica A 192:691–707 (1993).

  9. 9.

    A. Krönig and J. Jäckle, to be published (1993).

  10. 10.

    H. S. Wall,Analytic Theory of Continued Fractions (Chelsea, New York, 1967).

  11. 11.

    C. M. Bender and S. A. Orzag,Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

  12. 12.

    H. S. Carslaw and J. C. Jaeger,Conduction of Heat in Solids (Clarendon Press, Oxford, 1959).

  13. 13.

    H. Risken and H. D. Vollmer, Brownian motion in periodic potentials; Nonlinear response to external force,Z. Phys. B-Condensed Matter 33:297–305 (1979).

  14. 14.

    H. D. Vollmer and H. Risken, Distribution functions for Brownian motion of particles in a periodic potential driven by external force,Z. Phys. B-Condensed Matter 34:313–322 (1979).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Eisinger, S., Jäckle, J. Analytical approximations for the hierarchically constrained kinetic Ising chain. J Stat Phys 73, 643–670 (1993).

Download citation

Key words

  • Ising models
  • kinetic constraint
  • effective-medium approximation
  • mode-coupling approximation
  • cluster approximation
  • long-time behavior