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This study explores the spectral properties of stochastic processes, revealing a universal spectral feature related to relaxation rates. It also identifies a critical wave vector above which dynamics shift, with implications for quantum mechanics.

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Area of Science:

  • Statistical physics
  • Quantum mechanics
  • Non-equilibrium systems

Background:

  • Stochastic processes with finite propagation velocity are crucial for modeling various physical phenomena.
  • Generalized Poisson-Kac processes and Lévy walks represent key examples of regular and anomalous dynamics, respectively.
  • Understanding the spectral structure of evolution operators is vital for characterizing system dynamics.

Purpose of the Study:

  • To investigate the spectral structure of evolution operators for stochastic processes with finite propagation velocity.
  • To analyze the spectral features of Generalized Poisson-Kac processes and Lévy walks.
  • To explore the transition in relaxation dynamics at a critical wave vector and its quantum implications.

Main Methods:

  • Analysis of the spectral structure of evolution operators.
  • Explicit consideration of Generalized Poisson-Kac processes and Lévy walks.
  • Investigation of eigenvalue spectra and spectral dispersion curves.
  • Examination of stochastic states parametrized by velocity.

Main Results:

  • A generic spectral feature is the lower boundedness of the real part of the eigenvalue spectrum, limiting spectral dispersion.
  • This spectral feature physically represents an upper limit on the relaxation rate as a function of wave vector.
  • For Generalized Poisson-Kac processes with a continuum of states, a critical wave vector exists.
  • Above this critical wave vector, the point spectrum disappears, and essential spectrum governs relaxation dynamics.

Conclusions:

  • The spectral properties of these stochastic processes provide insights into relaxation dynamics and wave vector dependence.
  • The identified critical wave vector signifies a fundamental change in relaxation behavior.
  • The model serves as a foundational example for sub-quantum dynamics with hidden variables, extendable to quantum systems.