In this tutorial, three examples of stochastic systems are considered: a strongly damped oscillator, a weakly damped oscillator and an undamped oscillator (integrator) driven by noise. The evolution of these systems is characterized by the temporal correlation functions and spectral densities of their displacements, which are determined and discussed. Damped oscillators reach steady stochastic states. Their correlations are decreasing functions of the difference between the sample times, and their spectra have peaks near their resonance frequencies. An undamped oscillator never reaches a steady state. Its energy increases with time, and its spectrum is sharply peaked at low frequencies. The required mathematical methods and physical concepts are explained on a just-in-time basis, and some theoretical pitfalls are mentioned. The insights one gains from studies of oscillators can be applied to a wide variety of physical systems, such as atom and semiconductor lasers, which will be discussed in a subsequent tutorial.

中文翻译：

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随机系统：教程

在本教程中，考虑了随机系统的三个示例：强阻尼振荡器、弱阻尼振荡器和由噪声驱动的无阻尼振荡器（积分器）。这些系统的演化以它们位移的时间相关函数和谱密度为特征，它们被确定和讨论。阻尼振荡器达到稳定的随机状态。它们的相关性是采样时间差的递减函数，并且它们的光谱在它们的共振频率附近具有峰值。无阻尼振荡器永远不会达到稳定状态。它的能量随着时间的推移而增加，它的频谱在低频处急剧上升。及时解释了所需的数学方法和物理概念，并提到了一些理论陷阱。