Abstract
The critical slowing down of anisotropic magnets, binary mixtures, and systems undergoing structural transitions is interpreted in terms of suitable defined “clusters,” their growth, and their motions (cluster reactions, cluster diffusion, and cluster waves). Our previous studies of the Glauber model are extended considerably by numerical calculations, including the use of the cluster model of Reatto and Rastelli. The behavior of the relaxation function is very insensitive to the details of the models used. A scaling theory of nonlinear response is given, which is far more general than the cluster dynamics treatment. Two different cases occur:(i) at fixed “relative nonlinearity” the critical exponents agree with the corresponding exponents of linear response; (ii) if the initial state is held fixed, different exponents are found, however, which agree with predictions of Racz, and are consistent with Monte Carlo simulations of the nonlinear slowing down of the energy in kinetic Ising models.
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References
N. Ogita, A. Keda, T. Matsubara, H. Matsuda, and F. Yonezawa,J. Phys. Soc. Japan (Suppl.) 26:145 (1969).
L. P. Kadanoff, inCritical Phenomena, M. S. Green, ed., Academic, New York (1971), p. 118.
A. M. Gottlieb and P. Heller,Phys. Rev. B 3:3615 (1971).
G. J. Adriaenssens,Phys. Rev. B 12:5116 (1975); G. J. Adriaenssens, J. Bjorkstam, and J. Aikins,J. Mag. Resonance 7:99 (1972).
R. L. Armstrong and C. A. Martin,Phys. Rev. Lett. 35:294 (1975).
S. Aubry and R. Pick,Ferroelectrics 8:471 (1974).
S. Aubry, Thesis, CEN, Saclay (1974), unpublished.
J. A. Krumhansl and J. R. Schrieffer,Phys. Rev. B 11:3535 (1975).
T. Schneider and E. Stoll,J. Phys. C 8:283 (1975);Phys. Rev. B 13:1216 (1976).
B. J. Ackerson, C. M. Sorensen, R. C. Mockler, and W. J. O'Sullivan,Phys. Rev. Lett. 34:1371 (1975).
F. Schwabl, inAnharmonic Lattices, Structural Transitions and Melting, T. Riste, ed., Leiden, Nordhoff (1974).
K. Binder, D. Stauffer, and H. Müller-Krumbhaar,Phys. Rev. B 10:3853 (1974).
K. Binder, D. Stauffer, and H. Müller-Krumbhaar,Phys. Rev. B 12:5261 (1975).
K. Binder and D. Stauffer,Phys. Rev. Lett. 33:1006 (1974).
M. E. Fisher,Physics 3:255 (1967).
E. Stoll, K. Binder, and T. Schneider,Phys. Rev. B 6:2777 (1972); K. Binder and H. Müller-Krumbhaar,Phys. Rev. B 9:2328 (1974).
K. Binder,Ann. Phys. (N.Y.) 98:in press (1976).
H. Müller-Krumbhaar,Phys. Lett. 48A:459 (1974);50A:27 (1974).
J. W. Essam, inPhase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds., Academic, New York (1972), Vol. 2, p. 197.
L. P. Kadanoff,Physics 2:263 (1966).
K. G. Wilson and J. Kogut,Phys. Rep. 12C:75 (1974); K. G. Wilson,Rev. Mod. Phys. 41:773 (1975).
J. S. Langer,Physics 73:61 (1974).
A. I. Larkin and D. E. Khmelnitzkii,Soviet Phys.—JETP 29:1123 (1969).
M. E. Fisher,Rep. Prog. Phys. 30:615 (1967).
C. Domb and M. S. Green, eds.,Phase Transitions and Critical Phenomena, Academic, New York (1972), Vols. 1–6.
P. G. de Gennes, P. Lafore, and J. P. Millet,J. Phys. Chem. Solids 11:105 (1959).
K. Binder and D. Stauffer,J. Stat. Phys. 6:49 (1972); C. Domb, T. Schneider, and E. Stoll,J. Phys. A 8:L90 (1975).
C. Domb,J. Phys. A 9:283 (1976).
C. K. Majumdar,Phys. Rev. B 10:2857 (1974).
D. Stauffer, C. S. Kiang, and G. H. Walker,J. Stat. Phys. 3:323 (1971).
B. I. Halperin and P. C. Hohenberg,Phys. Rev. 177:952 (1969).
M. M. Bakri and D. Stauffer, preprint.
A. F. Andreev,Soviet Phys.—JETP 18:1415 (1964).
L. Reatto and E. Rastelli,J. Phys. C 5:2785 (1972).
K. Binder and D. Stauffer,Adv. Phys., in press (1976).
E. Stoll, K. Binder, and T. Schneider,Phys. Rev. B 8:3266 (1973).
L. van Hove,Physica 21:517 (1955).
D. Stauffer and K. Binder, in preparation.
B. I. Halperin, P. C. Hohenberg, and S. K. Ma,Phys. Rev. B 10:139 (1974).
T. L. Einstein and R. Cohen,Phys. Rev. B 7:1932 (1973).
T. P. Tseng, S. K. Feng, C. Cheng, and W. Band,J. Chem. Phys. 8:20 (1940); J. L. Katz, H. Saltsburg, and H. Reiss,J. Colloid Interface Sci. 21:560 (1966).
J. Frenkel,Kinetic Theory of Liquids, Dover, New York (1946), Chapter VII.
R. Zwanzig and N. K. Ailawadi,Phys. Rev. 182:280 (1969).
B. U. Felderhof,Rept. Math. Phys. 1:215 (1971);2:251 (1971).
R. Kretschmer,Diplomarbeit, Universität Saarbrücken (1975), unpublished.
L. Collatz,Eigenwertaufgaben mit technischen Anwendungen, Akademische Verlagsges., Leipzig (1963), p. 355.
A. Baumgärtner and K. Binder, in preparation.
R. Richtmeyer and K. W. Morton,Difference Methods for Initial Value Problems, Interscience, New York (1967), p. 190.
H. Yahata,J. Phys. Soc. Japan 30:657 (1971).
M. T. Hutchings, M. P. Schulhof, and H. J. Guggenheim,Phys. Rev. B 5:154 (1972).
M. Suzuki,Prog. Theor. Phys. 42, 1076 (1969);43, 882 (1970);Int. J. Magn. 1, 123 (1971).
Z. Racz,Phys. Rev. B 13:263 (1976); M. E. Fisher and Z. Racz, preprint.
M. F. Collins and Z. Racz,Phys. Rev. B 13:3074 (1976).
M. Suzuki, preprint.
K. Binder, inPhase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds., Academic, New York (1976), Vol. 5b, p. 1.
K. Binder and D. P. Landau,Phys. Rev. B 13:1140 (1976).
D. P. Landau, preprint.
H. C. Bolton and C. H. Johnson,Phys. Rev. B 13:3025 (1976).
M. F. Collins and H. C. Teh,Phys. Rev. Lett. 30, 781 (1973).
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Kretschmer, R., Binder, K. & Stauffer, D. Linear and nonlinear relaxation and cluster dynamics near critical points. J Stat Phys 15, 267–297 (1976). https://doi.org/10.1007/BF01023054
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DOI: https://doi.org/10.1007/BF01023054