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Eigenfunction statistics of complex systems: a common mathematical formulation.

Pragya Shukla1

  • 1Department of Physics, Indian Institute of Technology, Kharagpur, India.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 7, 2007
PubMed
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We developed a universal mathematical model for eigenfunction statistics in complex systems, using only system size and complexity. This contrasts with eigenvalue statistics, offering new ways to classify physical properties.

Area of Science:

  • Mathematical Physics
  • Quantum Mechanics
  • Statistical Mechanics

Background:

  • Eigenfunction and eigenvalue statistics are crucial for understanding complex systems.
  • Current models often lack a unified framework for diverse physical phenomena.

Purpose of the Study:

  • To derive a common mathematical formulation for eigenfunction statistics.
  • To identify universal scaling behaviors in complex systems.

Main Methods:

  • Developed a multiparametric probability density formulation.
  • Analyzed the dependence on system size and a complexity parameter.

Main Results:

  • Eigenfunction statistics depend on system size and complexity.
  • Eigenvalue statistics depend only on complexity, showing single-parameter scaling.

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  • Identified a common formulation for both, applicable across various complex systems.

    • Conclusions:

    • A unified mathematical framework for eigenfunction and eigenvalue statistics is possible.
    • This framework allows classification of systems into universality classes based on complexity.