Nonlinear Sciences – Chaotic Dynamics
Scientific paper
1996-08-26
Int. J. Mod. Phys. B 11 (1997) 805-849
Nonlinear Sciences
Chaotic Dynamics
44 pages, Postscript. The figures are included in low resolution only. A full version is available at http://www.physik.uni-
Scientific paper
10.1142/S0217979297000459
The mode-fluctuation distribution $P(W)$ is studied for chaotic as well as for non-chaotic quantum billiards. This statistic is discussed in the broader framework of the $E(k,L)$ functions being the probability of finding $k$ energy levels in a randomly chosen interval of length $L$, and 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 chaotic and non-chaotic systems. The vanishing of the normalized cumulants $C_k$, $k\geq 3$, implies a Gaussian behaviour of $P(W)$, which is realized in the case of chaotic systems, whereas non-chaotic systems display non-vanishing values for these cumulants leading to a non-Gaussian behaviour of $P(W)$. For some integrable systems there exist rigorous proofs of the non-Gaussian behaviour which are also discussed. Our numerical results and the rigorous results for integrable systems suggest that a clear fingerprint of chaotic systems is provided by a Gaussian distribution of the mode-fluctuation distribution $P(W)$.
Aurich Ralf
Bäcker Arnd
Steiner Frank
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