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

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.