Non Gaussian statistics in static and dynamic Galton boards

TL;DR

This paper investigates non-Gaussian stationary distributions in static and dynamic Galton boards, deriving exact and approximate solutions for the distributions under various conditions, revealing deviations from classical Gaussian behavior.

Contribution

It introduces new analytical solutions for non-Gaussian distributions in static and forced dynamic Galton boards, including exact cumulant generating functions and first-order approximations.

Findings

Static perturbations lead to non-Gaussian stationary distributions.

External forcing in dynamic Galton boards causes non-trivial distribution deviations.

Exact and approximate solutions characterize the distributions under different conditions.

Abstract

Perturbing the arrangements of pegs on a static Galton board can result in non-trivial stationary distributions, which in the continuum limit correspond to departure from regular gaussian behavior. Two such distributions are obtained. Further, the distributions generated for a dynamic galton board under external forcing in a general direction are obtained by solution of the corresponding stochastic differential equations. Exact cumulant generating functions for the distribution are presented for forcing in one dimension. An approximate expression, correct to first order in the forcing amplitude, is presented for the case of two dimensions. Both cases show nontrivial departures from the static gaussian solution.