The data of the article "The optimal incident angle during the saddle-shaped surface crossing process of Brownian particles"

The program simulates the behavior of particles based on Brownian motion in three-dimensional space and studies their dynamic characteristics on the saddle surface. The program first defines the saddle surface function, which describes the surface morphology of particles on a plane. Next, use numerical methods to generate high-resolution two-dimensional grids and calculate the corresponding saddle surface height values. In the simulation process, the displacement of particles in each time step is randomly generated in three directions through a normal distribution, while considering the small influence of initial velocity. Each simulation starts from a specific initial position and velocity direction, tracking the particle motion path and counting whether it crosses the saddle surface. In order to improve computational efficiency, the program adopts multi threading technology to parallel process simulations in different speed directions. The simulation results are adjusted using Gaussian distribution to ensure the smoothness of the probability distribution and exhibit peak characteristics within a specific angle range. Finally, the program saves the crossing probabilities corresponding to each velocity direction to a text file and generates graphics to display the distribution of crossing probabilities under different initial velocity directions. The entire process runs on standard computer equipment, utilizing scientific computing libraries for data processing and graphical display.