Corazza Giulio, Fadel Matteo
Laboratory for Computation and Visualization in Mathematics and Mechanics (LCVMM) Institute of Mathematics, Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland.
Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland.
Phys Rev E. 2020 Aug;102(2-1):022135. doi: 10.1103/PhysRevE.102.022135.
Path integrals play a crucial role in describing the dynamics of physical systems subject to classical or quantum noise. In fact, when correctly normalized, they express the probability of transition between two states of the system. In this work, we show a consistent approach to solve conditional and unconditional Euclidean (Wiener) Gaussian path integrals that allow us to compute transition probabilities in the semiclassical approximation from the solutions of a system of linear differential equations. Our method is particularly useful for investigating Fokker-Planck dynamics and the physics of stringlike objects such as polymers. To give some examples, we derive the time evolution of the d-dimensional Ornstein-Uhlenbeck process and of the Van der Pol oscillator driven by white noise. Moreover, we compute the end-to-end transition probability for a charged string at thermal equilibrium, when an external field is applied.
路径积分在描述受经典或量子噪声影响的物理系统动力学中起着至关重要的作用。实际上，经过正确归一化后，它们表示系统两个状态之间的跃迁概率。在这项工作中，我们展示了一种求解条件和无条件欧几里得（维纳）高斯路径积分的一致方法，这使我们能够根据线性微分方程组的解在半经典近似下计算跃迁概率。我们的方法对于研究福克 - 普朗克动力学以及诸如聚合物等类弦物体的物理性质特别有用。为了给出一些例子，我们推导了d维奥恩斯坦 - 乌伦贝克过程和由白噪声驱动的范德波尔振荡器的时间演化。此外，当施加外部场时，我们计算了处于热平衡的带电弦的端到端跃迁概率。
