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 ck(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 Ck, k ≥ 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).
[1]
J. I. Ramos,et al.
Hydrodynamic behavior and interacting particle systems
,
1990
.
[2]
Madan Lal Mehta,et al.
Random Matrices and the Statistical Theory of Energy Levels.
,
1970
.
[3]
R. Fisher.
The Advanced Theory of Statistics
,
1943,
Nature.
[4]
N. Wiener,et al.
Almost Periodic Functions
,
1932,
The Mathematical Gazette.
[5]
E. Mapother,et al.
Collected papers, vol. I
,
1925
.
[6]
G. L..
Collected Papers
,
1912,
Nature.
[7]
R. Schubert.
The Trace Formula and the Distribution of Eigenvalues of Schrödinger Operators on Manifolds all of whose Geodesics are Closed
,
1995
.
[8]
Michael V Berry,et al.
Some quantum-to-classical asymptotics
,
1991
.