The data of the article "Simulation of Brownian Particle Crossing a Two-Dimensional Saddle-Shaped Potential Energy Surface"

The program simulates the Brownian motion of a particle on a saddle surface. Firstly, a saddle surface function is defined, and the height value at each point is calculated by creating a high-resolution two-dimensional grid. The particle starts from a specified initial position, and each step displacement follows a normal distribution, representing the random movement process of the Brownian particle. In each iteration, the x and y coordinates of the particle are updated according to the randomly generated step length and are restricted within the specified grid range to ensure that the movement does not exceed the boundary. Meanwhile, the z value is updated based on the new coordinates to ensure that the particle always remains on the saddle-shaped surface. After the simulation was completed, to facilitate the observation and statistical analysis of the movement patterns, the particle trajectories were smoothed. A rolling mean filter with a window size of 120 was used to reduce noise and more accurately represent the actual movement of the particles. Ultimately, the particle trajectories were visualized in two ways: one was a 3D surface plot, where the trajectories were overlaid on the surface as red lines, and the other was a heatmap, which displayed the height distribution at different positions on the surface and superimposed the smoothed trajectories on it. The program utilized NumPy for numerical computations, Matplotlib for plotting, and Pandas for trajectory smoothing.