论文标题
多元Langevin方程中的确切电势
Exact potentials in multivariate Langevin equations
论文作者
论文摘要
由具有精确电势表现出直接动力学的多元Langevin方程控制的系统，但通常很难识别，因为在一般坐标变化之后，梯度流被映射的Jacobian矩阵掩盖。在这项工作中，提出了对一般非线性映射下Langevin方程的转换属性的详细分析。我们通过了解其差异几何特性来展示如何识别具有精确电势的系统。为了证明我们方法的力量，我们将其用于为非线性确定性和随机振荡的广泛研究模型提供精确的势。在选定的示例中，我们可视化确定的电位。我们的结果暗示了一系列可以从给定的确定性梯度系统中自一定义的广泛可解决的随机模型。
Systems governed by a multivariate Langevin equation featuring an exact potential exhibit straightforward dynamics but are often difficult to recognize because, after a general coordinate change, the gradient flow becomes obscured by the Jacobian matrix of the mapping. In this work, a detailed analysis of the transformation properties of Langevin equations under general nonlinear mappings is presented. We show how to identify systems with exact potentials by understanding their differential-geometric properties. To demonstrate the power of our method, we use it to derive exact potentials for broadly studied models of nonlinear deterministic and stochastic oscillations. In selected examples, we visualize the identified potentials. Our results imply a broad class of exactly solvable stochastic models which can be self-consistently defined from given deterministic gradient systems.
