MODE FLUCTUATIONS AS FINGERPRINTS OF CHAOTIC AND NONCHAOTIC SYSTEMS

The mode-fluctuation distribution P(W) is studied for chaotic as well as for non-chaotic quantum billiards. This statistic is discussed in t he broader framework of the E(k, L) functions being the probability of finding k energy levels in a randomly chosen interval of length L, an d the distribution of n(L), where n(L) is the number of levels in such an interval, and their cumulants c(k)(L). It is demonstrated that the cumulants provide a possible measure for the distinction between chao tic and non-chaotic systems. The vanishing of the normalized cumulants C-k, k greater than or equal to 3, implies a Gaussian behaviour of P( W), which is realized in the case of chaotic systems, whereas non-chao tic systems display non-vanishing values for these cumulants leading t o a non-Gaussian behaviour of P(W). For some integrable systems there exist rigorous proofs of the non-Gaussian behaviour which are also dis cussed. Our numerical results and the rigorous results for integrable systems suggest that a clear fingerprint of chaotic systems is provide d by a Gaussian distribution of the mode-fluctuation distribution P(W) .