Spectral distribution of a nonlinear oscillator performing Brownian motion in a double-well potential

Abstract

The spectral distribution Q(ω) of the coordinate fluctuations is studied for an oscillator performing Brownian motion in a double-well potential at low damping. The most detailed analysis is given for the Duffing oscillator with potential energy $\text{U(q) = −}\text{1}\text{2}\text{ω}{}^2{}_0\text{q}{}^2\text{+}\text{1}\text{4}\text{γq}{}^4$. The important features of Q(ω) are shown to be related to the slowing-down of the motion in the vicinity of the local potential maximum. In a certain range of the noise intensity, Q(ω) has three distinct peaks. They are due to fluctuational transitions between potential wells and to vibrations near the minima of U(q) and above the barrier. A typical feature of Q(ω) is the exponential tail in the region ω < ω₀ passing into a plateau at still smaller ω (but ω å Γ). The plateau depends on the friction coefficient Γ nonanalytically (as √Γ).