Statistical Theory of the Energy Levels of Complex Systems. I

Journal Article · Mon Jan 01 04:00:00 EST 1962 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.1703773· OSTI ID:4801180

Dyson, Freeman J.

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 ensernble 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 symmetry. The probability-distributions for the energy levels are calculated. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistscal mechanics of a one-dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group-theoretical argu- ,;-ments, the conjecture is proved for the three temperatures . which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. (auth)

• Research Organization: Inst. for Advanced Study, Princeton, N.J.
• Sponsoring Organization: USDOE
• NSA Number: NSA-16-012473
• OSTI ID: 4801180
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 1 Vol. 3; ISSN JMAPAQ; ISSN 0022-2488
• Publisher: American Institute of Physics (AIP)
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

DISTRIBUTION
EIGENVALUES
ENERGY LEVELS
GASES
GROUP THEORY
INTERACTIONS
MATHEMATICS
NUMERICALS
PHYSICS
PROBABILITY THEORY
STATISTICS
TEMPERATURE
THERMODYNAMICS