论文标题
随机非线性振荡器的示意扰动理论
Diagrammatic perturbation theory for Stochastic nonlinear oscillators
论文作者
论文摘要
我们考虑随机驱动的一维非线性振荡器$ \ ddot {x}+2γ\ dot {x}+ω^2_0 x+λx^3 = f（t）$，其中f（t）是高斯噪声，对于工作的大部分，delta是相关的（白噪声）。我们在基于系统的Feynman图扰动理论中构建频率空间中的线性响应函数。与其他物理领域一样，这种扩展的特征是图中的环数。这使我们可以证明，阻尼系数以$ O（λ^2）$（这是两个环订单）获得了校正。更重要的是，它导致了数值小但概念上有趣的发现，即响应是探测随机系统的频率的函数。该方法易于推广到不同种类的非线性，并在上述方程中替换了$μx^2 $中的非线性项，我们可以讨论噪声驱动的逃脱逃脱的问题。如果我们在立方非线性情况下添加周期性强迫，那么我们发现响应函数可以与噪声强度和周期性驱动的幅度共同成比例。为了治疗扰动理论中的随机卡皮扎问题，我们发现有彩色噪声是有必要的。
We consider the stochastically driven one dimensional nonlinear oscillator $\ddot{x}+2Γ\dot{x}+ω^2_0 x+λx^3 = f(t)$ 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 $O(λ^2)$ 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 $μx^2$ , 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.
