Generalization of the Mean-Field Method for Power-Law Distributions

Recently much attention has been paid to the nonextensive canonical distributions: the αfamilies. Such distributions have been found in many real-world systems such as fully developed turbulence and financial markets. In this paper, a generalized mean-field method to approximate the expectations of the α-families is proposed. We calculate the α′-projection of a probability distribution to find that the computational complexity to approximate the expectations is greatly reduced with a proper choice of the projection-index α′. We apply this method to a simple binary-state system and compare the results with direct numerical calculations.