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Renormalization group and critical phenomena. I: Renormalization group and the Kadanoff scaling picture. (English) Zbl 1236.82017
Summary: The Kadanoff theory of scaling near the critical point for an Ising ferromagnet is cast in differential form. The resulting differential equations are an example of the differential equations of the renormalization group. It is shown that the Widom-Kadanoff scaling laws arise naturally from these differential equations if the coefficients in the equations are analytic at the critical point. A generalization of the Kadanoff scaling picture involving an “irrelevant” variable is considered; in this case the scaling laws result from the renormalization-group equations only if the solution of the equations goes asymptotically to a fixed point.
For part II, cf. Phys. Rev. B (3) 4, No. 9, 3184–3205 (1971; Zbl 1236.82016).

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics
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[1] M. Fisher, Rept. Progr. Phys. 30 pp 731– (1967)
[2] L. P. Kadanoff, Rev. Mod. Phys. 39 pp 395– (1967)
[3] M. Gell-Mann, Phys. Rev. 95 pp 1300– (1954) · Zbl 0057.21401
[4] E. C. G. Stueckelberg, Helv. Phys. Acta 26 pp 499– (1953)
[5] N. N. Bogoliubov, in: Introduction to the Theory of Quantized Fields (1959)
[6] K. Wilson, Phys. Rev. D 3 pp 1818– (1971)
[7] C. DiCastro, Phys. Letters 29A pp 322– (1969)
[8] A. A. Migdal, Zh. Eksperim. i Teor. Fiz. 59 pp 1015– (1970)
[9] K. Wilson, Phys. Rev. B 4 pp 3184– (1971) · Zbl 1236.82016
[10] L. P. Kadanoff, Physics 2 pp 263– (1966)
[11] D. Jasnow, Phys. Rev. 176 pp 739– (1968)
[12] E. Riedel, Z. Physik 225 pp 195– (1969)
[13] E. Riedel, Phys. Rev. Letters 24 pp 730– (1970)
[14] E. Riedel, Phys. Rev. Letters 24 pp 930– (1970)
[15] N. Minorsky, in: Nonlinear Oscillations (1962)
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