Journal of Statistical Physics

, Volume 78, Issue 3–4, pp 893–916 | Cite as

Relaxation times in a finite Ising system with random impurities

  • K. Oerding


Finite-size scaling effects of the Ising model with quenched random impurities are studied, focusing on critical dynamics. In contrast to the pure Ising model, disordered systems are characterized by continuous relaxation time spectra. Dynamic field theory is applied to compute the spectral densities of the magnetizationM(t) and ofM2(t). In addition, universal cumulant ratios are calculated to second order in ε1/4, where ε=4−d andd<4 denotes the spatial dimension.

Key Words

Dynamic critical phenomena disordered spin systems Ising model finite size scaling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. O. Mayer, Critical exponents of the dilute Ising model from four-loop expansions,J. Phys. A 22:2815–2823 (1989).Google Scholar
  2. 2.
    D. J. W. Geldart and K. De'Bell, Logarithmic corrections for dilute uniaxial ferromagnets at the critical dimension,J. Stat. Phys. 73:409–414 (1993).Google Scholar
  3. 3.
    H.-O. Heuer, Monte Carlo simulation of strongly disordered Ising ferromagnets,Phys. Rev. B 42:6476–6484 (1990).Google Scholar
  4. 4.
    H.-O. Heuer, Critical crossover phenomena in disordered Ising systems,J. Phys. A 26:L333-L339 (1993).Google Scholar
  5. 5.
    H.-O. Heuer, Dynamic scaling of disordered Ising systems,J. Phys. A 26:L341-L346 (1993).Google Scholar
  6. 6.
    E. Sengespeick, Feldtheorie des kritischen Verhaltens von ungeordneten Ising-Systemen, Diploma thesis, Düsseldorf (1994); E. Sengespeick, H. K. Janssen, and K. Oerding, to be published.Google Scholar
  7. 7.
    M. E. Fisher, The theory of critical point singularities, inCritical Phenomena, M. S. Green, ed. (Academic Press, New York, 1971), pp. 1–99.Google Scholar
  8. 8.
    E. Brézin, and J. Zinn-Justin, Finite size effects in phase transitions,Nucl. Phys. B 257 [FS14]:867–893 (1985).Google Scholar
  9. 9.
    H. W. Diehl, Finite size effects in critical dynamics and the renormalization group.Z. Phys. B 66:211–218 (1987).Google Scholar
  10. 10.
    Y. Y. Goldschmidt, Finite size scaling effects in dynamics,Nucl. Phys. B 280 [FS18]: 340–354 (1987).Google Scholar
  11. 11.
    J. C. Niel and J. Zinn-Justin, Finite size effects in critical dynamics,Nucl. Phys. B 280 [FS18]:355–384 (1987).Google Scholar
  12. 12.
    H. K. Janssen, B. Schaub, and B. Schmittmann, Finite size scaling for directed percolation and related stochastic evolution process,Z. Phys. B 71:377–385 (1988).Google Scholar
  13. 13.
    H. K. Janssen, B. Schaub, and B. Schmittmann, The general epidemic process in a finite environment.J. Phys. A 21:L427-L434 (1988).Google Scholar
  14. 14.
    H. K. Janssen, On a Lagrangian for classical field dynamics and renormalization group calculations of dynamical properties,Z. Phys. B 23:377–380 (1976); C. De Dominicis, Techniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques,J. Phys. (Paris)Coll. C1:247–253 (1976).Google Scholar
  15. 15.
    C. De Dominicis and L. Peliti, Field-theory renormalization and critical dynamics aboveT c: Helium, antiferromagnets, and liquid-gas systems,Phys. Rev. B 18:353–376 (1978).Google Scholar
  16. 16.
    H. K. Janssen, On the renormalized field theory of nonlinear critical relaxation, inFrom Phase Transitions to Chaos, Topics in Modern Statistical Physics, G. Györgyi, I. Kondor, L. Sasvári, and T. Tél, eds. (World Scientific, Singapore, 1992), p. 68–91.Google Scholar
  17. 17.
    A. B. Harris, Effect of random defects on the critical behavior of Ising models,J. Phys. C 7:1671–1692 (1974); A. B. Harris and T. C. Lubensky, Renormalization-group approach to the critical behavior of random-spin models,Phys. Rev. Lett. 33:1540–1543 (1974).Google Scholar
  18. 18.
    C. De Dominicis, Dynamics as a substitute for replicas in systems with quenched random impurities,Phys. Rev. B 18:4913–4919 (1978).Google Scholar
  19. 19.
    I. D. Lawrie and V. V. Prudnikov, Static and dynamic properties of systems with extended defects: Two-loop approximation,J. Phys. C 17:1655–1668 (1984).Google Scholar
  20. 20.
    I. H. Sneddon,The Use of Integral Transforms (McGraw-Hill, New York, 1972).Google Scholar
  21. 21.
    K. Binder, Finite size scaling analysis of Ising model block distribution functions,Z. Phys. B 43:119–140 (1981).Google Scholar
  22. 22.
    Y. Y. Goldschmidt, Dynamical relaxation in finite size system.Nucl. Phys. B 285[FS19]:519–534 (1987).Google Scholar
  23. 23.
    H. Dekker and N. G. van Kampen, Eigenvalues of a diffusion process with a critical point,Phys. Lett. A 73:374–376 (1979).Google Scholar
  24. 24.
    R. W. Daniels,An Introduction to Numerical Methods and Optimization Techniques (Elsevier North-Holland, New York, 1978).Google Scholar
  25. 25.
    S. Wolfram,Mathematica (Addison-Wesley, Redwood City, California, 1991).Google Scholar
  26. 26.
    R. B. Griffiths, Nonanalytic behavior above the critical point in a random Ising ferromagnetPhys. Rev. Lett. 23:17–19 (1969); J. L. Cardy and A. J. McKane, Field theoretic approach to the study of Yang-Lee and Griffiths singularities in the randomly diluted Ising model,Nucl. Phys. B 257[FS14]:383–396 (1985).Google Scholar
  27. 27.
    H. K. Janssen, B. Schaub, and B. Schmittmann, New universal short-time scaling behavior of critical relaxation processes,Z. Phys. B 73:539–549 (1989).Google Scholar
  28. 28.
    H. W. Diehl and U. Ritschel, Dynamical relaxation and universal short-time behavior in finite systems,J. Stat. Phys. 73:1–20 (1993).Google Scholar
  29. 29.
    J. Zinn-Justin, The principles of instanton calculus, inLectures at Les Houches Summer School, J.-B. Zuber and R. Stora, eds. (North-Holland, Amsterdam, 1984), p. 39.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • K. Oerding
    • 1
  1. 1.Institut für Theoretische Physik IIIHeinrich-Heine-UniversitätDüsseldorfGermany

Personalised recommendations