THE EFFECT OF WALL REBOUNDS ON THE EXIT DISTRIBUTION FUNCTION IN GALTON BOARD MODELS

In an ideal Galton board model, the exit distribution function represents the solution to a one-dimensional diffusion equation, providing a means to verify the correctness of the derived distribution. However, when wall rebounds are incorporated, the exit distribution function obtained through the maximum entropy principle fails to satisfy this diffusion equation. To rigorously incorporate wall rebound effects, we employ a periodic folding scheme that yields an exit distribution function satisfying the one-dimensional diffusion equation. We demonstrate that the maximum entropy-based exit distribution function serves as a unimodal approximation of the periodic folding scheme. Numerical experiments using Monte Carlo method reveal that for exit distribution functions incorporating wall rebounds, a three-peak approximation derived from the periodic folding scheme exhibits superior expressive capability compared to the unimodal approximation. Furthermore, as the number of peg layers increases, the exit distribution function undergoes a gradual transition from a normal distribution toward a uniform distribution. Using the unimodal distribution as a representative case, we present a methodology for identifying this transition point.