MEAN FIELD FLUCTUATION OF A NETWORK OF CHAOTIC ELEMENTS - REMAINING FLUCTUATION AND CORRELATION IN THE LARGE SIZE LIMIT

Fluctuation of the mean field is studied for a network of chaotic elements with the use of globally coupled maps. As the size N is increased, the variance of the mean field decreases up to a crossover size N(c) but then stays constant, where the remaining variance is roughly proportional to 1/N(c). The distribution of the mean field approaches a Gaussian form up to N(c), but keeps some deviation for N > N(c). The breakdown of the law of large numbers implies some hidden coherence among elements, confirmed by finite mutual information and Lyapunov spectra. The breakdown is also supported by the instability of the self-consistent solution for the Perron-Frobenius operator. The anomalous effect of additional noise to our system is also reported. Results from other globally coupled maps suggest that the law of large numbers is generally broken in a turbulent phase with local stretching and contraction, but is satisfied for a map with stretching only.