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Journal of the Physical Society of Japan

Statistical Mechanical Theory of Cooperative Phenomena.I. General Theory of Fluctuations, Coherent Anomalies and Scaling Exponents with Simple Applications to Critical Phenomena

Masuo Suzuki

Published Dec 15, 1986• 210 citations
Publisher

Abstract

A general method is proposed to study non-classical scaling behaviors of cooperative systems. This is based on the appearance of a "coherent anomaly" in mean-field-type or classical approximations of cooperative systems. The degree of approximation is analytically continued to construct a parameterized expression of a whole set of approximations, using Kubo's linear response theory. General relations between scaling (or critical) exponents and "coherent-anomaly exponents" are derived using the theory of envelope and are also obtained from a finite-degree-of-approximation scaling, which is inspired by Fisher's finite-size scaling. The present coherent-anomaly method (CAM) is promising for calculating analytically non-classical scaling exponents in cooperative phenomena. A general scheme of CAM in critical phenomena is presented. It is shown that even the combination of the Weiss and Bethe approximations gives rather good estimates of static and dynamic critical exponents.

DOI
10.1143/jpsj.55.4205

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Journal of the Physical Society of Japan•Nov 15, 1988•26 citations
A general method is proposed to study non-classical scaling behaviors of cooperative systems. This is based on the appearance of a "coherent anomaly" in mean-field-type or classical approximations of cooperative systems. The degree of approximation is analytically continued to construct a parameterized expression of a whole set of approximations, using Kubo's linear response theory. General relations between scaling (or critical) exponents and "coherent-anomaly exponents" are derived using the theory of envelope and are also obtained from a finite-degree-of-approximation scaling, which is inspired by Fisher's finite-size scaling. The present coherent-anomaly method (CAM) is promising for calculating analytically non-classical scaling exponents in cooperative phenomena. A general scheme of CAM in critical phenomena is presented. It is shown that even the combination of the Weiss and Bethe approximations gives rather good estimates of static and dynamic critical exponents.
A coherent anomaly method (CAM) is formulated for critical phenomena on the basis of the general idea of CAM for cooperative phenomena proposed by Suzuki. The coherent anomalies of response functions are derived phenomenologically using Kubo's linear response theory and Fisher's scaling form of correlation functions. This yields the relation between the coherent anomaly exponents and non-classical critical exponents. The convergence of the CAM critical points is proven rigorously. Some explicit applications of the CAM theory based on systematic cluster-mean-field approximations and on the mean-field transfer-matrix method are presented to show how useful the CAM is in studying critical phenomena. The CAM theory of critical dynamics is also formulated to give the dynamical critical exponent Δ ≃1.84 in the two-dimensional kinetic Ising model.
The coherent-anomaly method (CAM) is applied to the kinetic Ising model. Dynamical cluster-mean-field approximations are formulated to obtain a series of the dynamical mean-field critical coefficients. The coherent-anomaly scaling relations are derived on the basis of the scaling form of the generating function in nonequilibrium systems to estimate the exponent Δ of the critical slowing down. The dynamical critical exponent is estimated as Δ⋍2.15(±0.02){\varDelta}{\backsimeq}2.15({\pm}0.02)Δ⋍2.15(±0.02) for the kinetic Ising model on the two-dimensional triangular lattice.
A general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.
A new scheme of the coherent-anomaly method (CAM) is proposed to study critical phenomena in the models for which a mean-field description gives spurious first-order phase transition. A canonical series of mean-field-type approximations are constructed so that the spurious discontinuity should vanish asymptotically as the approximate critical temperature approachs the true value. The true value of the critical exponents β and γ are related to the coherent-anomaly exponents defined among the classical approximations. The formulation is demonstrated in the two-dimensional q -state Potts models for q =3 and 4. The result shows that the present method enables us to estimate the critical exponents with high accuracy by using the date of the cluster-mean-field approximations.
A general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.
The coherent-anomaly scaling is investigated for the approximations derived on the basis of the cluster-variation method (CVM). All of the mean-field critical coefficients are calculated in the Weiss, the Bethe and the Kikuchi approximations for the three-dimensional Ising model on the simple cubic lattice. The dynamical coefficients of the relaxation time are also obtained in the corresponding approximations for the kinetic Ising model. The non-classical values of critical exponents as well as the true value of critical temperature are estimated by extrapolating these approximants by the coherent-anomaly method. The results suggest that the approximations by the CVM possess the property of coherent-anomaly scaling.