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Extreme statistics of complex random and quantum chaotic states
Arul Lakshminarayan1, Steven Tomsovic, Oriol Bohigas
1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany.
Physical Review Letters
|March 21, 2008
Summary
Complex random states exhibit statistical properties aligning with random matrix theory. Analytic formulas derived for extreme values show distinct convergence patterns for maximum and minimum intensities, validated in chaotic quantum systems.
Area of Science:
- Quantum mechanics
- Statistical physics
- Random matrix theory
Background:
- Complex random states share statistical properties with Gaussian and circular unitary ensembles in random matrix theory.
- Despite normalization constraints causing correlations, analytic formulas for extreme value statistics are derivable.
Purpose of the Study:
- To derive compact analytic formulas for the statistical properties of extreme values of complex random states.
- To investigate the applicability of random matrix theory to chaotic quantum systems by analyzing extreme eigenfunction statistics.
Main Methods:
- Derivation of analytic formulas for extreme value statistics of complex random states.
- Calculation of extreme eigenfunction statistics for the standard map in a fully chaotic regime.
Main Results:
- Maximum intensity statistics approach the Gumbel distribution, while minimum intensity statistics rapidly approach the Weibull distribution.
- Calculated extreme eigenfunction statistics for the chaotic standard map are consistent with finite-N formulas from random matrix theory.
Conclusions:
- Analytic formulas for extreme value statistics of complex random states are robust across dimensions.
- Random matrix theory provides a valid framework for understanding the statistical behavior of chaotic quantum systems.
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