Simulation and Analysis of Two Toy Models

The matching problem and the distribution law of Galton boards with interactions are studied in this paper. The general matching problem appeals at many scenarios, such as the reaction rate of molecules and the hailing rate of ride-hailing drivers. The Galton board is often used in the classroom as a demonstration experiment for the probability distribution of independent events. The two problems are mathematically modeled and numerically simulated. The expected value of matching rate is derived as an analytical solution of the partial differential equation and confirmed by simulation experiments. The interactions were introduced to Galton boards via two parameters in the toy model, which lead to Gaussian distributions of independent events cannot fit the experimental data well. Instead, 'quantum' Fermi-Dirac distributions unexpectedly conforms to simulation experiments. The exclusivity between particles leads to negative Chemical potential in the distribution function, and the temperature parameter increases with the interaction intensity $\alpha$ and flow rate $N_{sm}$. The relations between parameters can be expressed as a conjecture formula within large parameters range.