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Self-consistent Born theory for the Duffing oscillator
1Instituto Balseiro (UNCUYO), Centro Atómico Bariloche (CNEA), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), C1425FQB Buenos Aires, Argentina and , - , 8400 San Carlos de Bariloche, Argentina.
Physical Review. E
|May 16, 2026
Summary
We developed a self-consistent Born approximation (SCBA) for nonlinear oscillators driven by noise. This method accurately predicts system behavior, outperforming standard approximations, especially for large amplitudes.
Area of Science:
- Nonlinear dynamics
- Statistical physics
- Field theory
Background:
- Nonlinear oscillators driven by stochastic forces are common in physics.
- Standard perturbative methods often fail to accurately describe these systems, especially in the nonlinear regime.
Purpose of the Study:
- To develop a robust analytical framework for studying nonlinear oscillators under stochastic driving.
- To accurately predict system observables like mean-square displacement and renormalized frequency.
Main Methods:
- Development of a self-consistent Born approximation (SCBA) within a field-theoretic framework.
- Construction of a self-consistent modal mean-field solution analogous to particle field theory.
- Incorporation of dynamic correlations between noise-activated internal Fourier modes (NAIFMs).
Main Results:
- The SCBA successfully renormalizes the oscillator's natural frequency and captures amplitude-frequency dependence.
- Static and dynamic correlations among NAIFMs are effectively included.
- The SCBA accurately reproduces the mean-square displacement (MSD) and renormalized frequency, agreeing well with simulations, unlike perturbative expansions.
Conclusions:
- The SCBA provides a reliable analytical method for nonlinear oscillators subjected to stochastic driving.
- This approach is highly relevant for understanding micro- and nanomechanical resonators.
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