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Spectrum of stochastic evolution operators: local matrix representation approach.

P Cvitanović1, N Søndergaard, G Palla

  • 1Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
Summary

This study introduces a novel matrix method to analyze nonlinear stochastic flows, simplifying spectral calculations. The approach offers a more accessible way to compute perturbative corrections for complex systems.

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Area of Science:

  • Nonlinear dynamics
  • Stochastic processes
  • Quantum mechanics
  • Mathematical physics

Background:

  • Analyzing the spectrum of nonlinear stochastic flows is computationally challenging.
  • Existing methods like Feynman diagram perturbation theory can be complex to implement.
  • Semiclassical approximations are crucial for understanding spectral properties in quantum and stochastic systems.

Purpose of the Study:

  • To develop a more efficient matrix-based method for computing the spectrum of nonlinear stochastic flows.
  • To formulate a perturbative expansion for the spectrum in the weak noise limit.
  • To enable computation of perturbative corrections beyond current attainable orders.

Main Methods:

  • Utilizing a matrix representation of the evolution operator for nonlinear stochastic flows with additive noise.

Related Experiment Videos

  • Developing a perturbative expansion based on local matrix representations centered on classical periodic orbits.
  • Comparing the computational ease with standard Feynman diagram perturbation theory.

    • Main Results:

    • Successfully computed the spectrum of the nonlinear stochastic flow using the matrix representation.
    • Formulated a perturbative expansion for the spectrum in the weak noise limit.
    • Achieved computation of perturbative corrections to a stochastic analog of the Gutzwiller semiclassical spectral determinant to several higher orders.

    Conclusions:

    • The proposed matrix representation offers a computationally advantageous framework for spectral analysis of nonlinear stochastic systems.
    • This method facilitates the calculation of higher-order perturbative corrections, advancing understanding in stochastic and quantum mechanics.
    • The findings provide a new tool for exploring semiclassical spectral properties in complex dynamical systems.