Diagrammatic perturbation theory for Stochastic nonlinear oscillators

TL;DR

This paper develops a diagrammatic perturbation theory for stochastic nonlinear oscillators, revealing how noise and nonlinearity influence response functions and damping, with potential applications to escape problems and driven systems.

Contribution

It introduces a systematic Feynman diagram approach to analyze stochastic nonlinear oscillators, including corrections to damping and response functions at two-loop order, and extends to colored noise and driven systems.

Findings

Damping coefficient receives a two-loop correction.

Response depends on the probing frequency.

Response can be influenced by noise and periodic drive.

Abstract

We consider the stochastically driven one dimensional nonlinear oscillator $\ddot{x}+2\Gamma\dot{x}+\omega^2_0 x+\lambda x^3 = f(t)$ where f(t) is a Gaussian noise which, for the bulk of the work, is delta correlated (white noise). We construct the linear response function in frequency space in a systematic Feynman diagram-based perturbation theory. As in other areas of physics, this expansion is characterized by the number of loops in the diagram. This allows us to show that the damping coefficient acquires a correction at $O(\lambda^2)$ which is the two loop order. More importantly, it leads to the numerically small but conceptually interesting finding that the response is a function of the frequency at which a stochastic system is probed. The method is easily generalizable to different kinds of nonlinearity and replacing the nonlinear term in the above equation by $\mu x^2$ , we can discuss the issue of noise driven escape from a potential well. If we add a periodic forcing to the cubic nonlinearity situation, then we find that the response function can have a contribution jointly proportional to the strength of the noise and the amplitude of the periodic drive. To treat the stochastic Kapitza problem in perturbation theory we find that it is necessary to have a coloured noise.