论文标题
由水平随机力驱动的倒摆：永无止境的轨迹和超对称性的统计数据
Inverted pendulum driven by a horizontal random force: statistics of the never-falling trajectory and supersymmetry
论文作者
论文摘要
我们研究倒置的随机动力学在水平方向上受到随机力的影响（惠特尼的问题）。在整个时间轴上考虑了问题，该问题承认了一个独特的解决方案，该解决方案始终保留在上半平面。我们提出了这种永无止境的轨迹的统计描述问题，并通过假设白噪声驾驶的现场理论技术解决了问题。在我们基于巴黎和酸拉萨的超对称形式的方法中，以相应的转移 - 马trix hamiltonian的零模式表达了永无止境的轨迹的统计特性。新兴的数学结构与fokker-Planck方程的数学结构相似，但是它是为概率分布函数的“平方根”编写的。我们对非坠落轨迹的统计结果的结果与随机摆动方程的直接数值模拟完全吻合。在强驾驶的极限（无重力）的限制下，我们获得了一个精确的分析解决方案，该解决方案对于摆锤角度及其速度的瞬时关节概率分布函数。
We study stochastic dynamics of an inverted pendulum subject to a random force in the horizontal direction (Whitney's problem). Considered on the entire time axis, the problem admits a unique solution that always remains in the upper half plane. We formulate the problem of statistical description of this never-falling trajectory and solve it by a field-theoretical technique assuming a white-noise driving. In our approach based on the supersymmetric formalism of Parisi and Sourlas, statistic properties of the never-falling trajectory are expressed in terms of the zero mode of the corresponding transfer-matrix Hamiltonian. The emerging mathematical structure is similar to that of the Fokker-Planck equation, which however is written for the "square root" of the probability distribution function. Our results for the statistics of the non-falling trajectory are in perfect agreement with direct numerical simulations of the stochastic pendulum equation. In the limit of strong driving (no gravitation), we obtain an exact analytical solution for the instantaneous joint probability distribution function of the pendulum's angle and its velocity.
