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Nonspectral relaxation in one dimensional Ornstein-Uhlenbeck processes.
R Toenjes1, I M Sokolov2, E B Postnikov3
1Institute of Physics and Astronomy, Potsdam University, Potsdam-Golm 14476, Germany.
Physical Review Letters
|August 29, 2014
Summary
System relaxation rates usually depend on the operator spectrum, not initial conditions. However, some initial conditions lead to non-spectral relaxation, especially in generalized Ornstein-Uhlenbeck processes (OUPs) with Lévy noise.
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.
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