# Statistical Theory of the Energy Levels of Complex Systems. I

@article{Dyson1962StatisticalTO, title={Statistical Theory of the Energy Levels of Complex Systems. I}, author={Freeman J. Dyson}, journal={Journal of Mathematical Physics}, year={1962}, volume={3}, pages={140-156} }

New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time‐reversal invariance is violated, and the symplectic ensemble applies only to odd‐spin systems without rotational…

## 1,718 Citations

Orthogonal polynomial ensembles in probability theory

- Mathematics
- 2005

We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an {it orthogonal polynomial ensemble}. The most prominent…

Statistical theory of energy levels and random matrices in physics

- Mathematics
- 1973

In this paper the physical aspects of the statistical theory of the energy levels of complex physical systems and their relation to the mathematical theory of random matrices are discussed. After a…

Eigenvalue correlations in the circular ensembles

- Physics
- 1991

Dyson (1972) introduced two types of Brownian-motion ensembles of random matrices for studying approximate symmetries in complex quantum systems. The magnitude of symmetry breaking plays the role of…

Topological field theory approach to intermediate statistics

- Mathematics
- 2021

Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are…

Can quantum many-body systems behave as strongly chaotic, being completely integrable ?

- Physics
- 2019

We study the paradigmatic Lieb-Liniger (LL) model belonging to the class of integrable quantum many-body systems, by considering its statistical properties in the many-body Hilbert space. We…

Some studies on arithmetical chaos in classical and quantum mechanics

- Physics, Mathematics
- 1993

Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of…

Thermodynamic relations of the Hermitian matrix ensembles

- Mathematics
- 1997

Applying the Coulomb fluid approach to the Hermitian random matrix ensembles, universal derivatives of the free energy for a system of N logarithmically repelling classical particles under the…

Circulant L-ensembles in the thermodynamic limit

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2021

L-ensembles are a class of determinantal point processes which can be viewed as a statistical mechanical systems in the grand canonical ensemble. Circulant L-ensembles are the subclass which are…

Random matrix theory in semiclassical quantum mechanics of chaotic systems

- Physics
- 1988

The statistical properties of the spectrum of systems which have a chaotic classical limit have been found to be similar to those of random matrix ensembles. The author explains this correspondence,…

## References

SHOWING 1-10 OF 22 REFERENCES

The classical groups : their invariants and representations

- Mathematics
- 1940

In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from…

Extension of the Shell Model for Heavy Spherical Nuclei

- Physics
- 1960

The Bardeen-Bogoliubov-Belyaev treatment of the pairing correlations is applied to spherical nuclei with a general nuclear force. The interaction between quasi-particles is treated by the method of…

"Repulsion of Energy Levels" in Complex Atomic Spectra

- Physics
- 1960

It is shown that "repulsion of energy levels" of the same symmetry type occurs in complex atomic spectra. Thus, for the elements Hf, Ta, W, Re, Os, and Ir, for which the spin-dependent forces are…

Theory of Lie Groups

- Mathematics
- 1946

This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this…

Proof of a Conjecture by Dyson in the Statistical Theory of Energy Levels

- Mathematics
- 1962

A conjectured identity relating the statistical properties of two types of ensembles occurring in a statistical theory of the distribution of energy levels in nuclei and other complex systems is…

The Intrinsic Parity of Elementary Particles

- Physics
- 1952

The limitations to the concept of parity of quantum-mechanical states and, in particular, of intrinsic parity of elementary particles are discussed. These limitations are shown to follow from…

SLOW NEUTRON RESONANCE SPECTROSCOPY I, U238

- Physics
- 1960

The results of time-of-flight measurements of U/sup 238/ resonances in the region 90 to 1300 ev are presented and resonance parameters for levels up to 1000 ev are obtained. Neutron widths for the 55…

Slow Neutron Resonance Spectroscopy. II. Ag, Au, Ta

- Physics
- 1960

The results of time-of-flight measurements of silver, gold, and tantalum resonance parameters are presented. Neutron widths are given for 79 levels in silver to 728 ev, 55 levels in gold to 940 ev,…