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Area of Science:

  • Statistical physics
  • Non-equilibrium dynamics
  • Stochastic processes

Background:

  • Dissipative systems typically relax to equilibrium via multiexponential patterns.
  • Relaxation rates are derived from the Hermitian operator spectrum, independent of initial conditions.
  • Similarity transformations map Fokker-Planck operators to Hermitian ones.

Purpose of the Study:

  • Investigate relaxation patterns for initial conditions leading to functions growing at infinity.
  • Determine conditions under which relaxation rates belong to the Hermitian spectrum.
  • Analyze deviations from standard relaxation in Ornstein-Uhlenbeck processes (OUPs).

Main Methods:

  • Exact solutions for Gaussian and generalized Lévy Ornstein-Uhlenbeck processes (OUPs).
  • Analysis of similarity transformations of Fokker-Planck operators.
  • Comparison of spectral and non-spectral relaxation behaviors.

Main Results:

  • Relaxation rates align with the Hermitian spectrum only if initial conditions fall within the stable distribution's domain of attraction.
  • Non-spectral relaxation occurs for initial conditions mapped to functions growing at infinity.
  • Generalized OUPs driven by Lévy noise commonly exhibit non-spectral relaxation.

Conclusions:

  • The assumption of initial-condition-independent relaxation rates is not universally valid.
  • Lévy noise in generalized OUPs fundamentally alters relaxation dynamics.
  • Understanding the domain of attraction is crucial for predicting system relaxation behavior.