Creating kappa-like distributions from a Galton board

The Galton board is a classical device demonstrating geometrically the formation of normal distributions. In view of long-range interactions, governed by nonextensive statistics, the concept of the Galton board is extended providing numerically the corresponding power-law distributions. Within nonextensive statistics kappa-distributions, which are linked to qq-Gaussians, are a consequence from entropy generalization. In this way the transition from normal distributions to kappa-like distributions (where kappa is the entropic index) is available within a Galton board concept where both the generalized distributions for positive and negative kappa-like values are reproduced. Based on two similar numerical approaches it is shown how positive kappa-like exponents are related to memory effects and negative kappa-like exponents to enhanced interactions of increased order in the system, as compared to the normal distribution case.

Research highlights► The Galton board shows the formation of normal distributions. ► We extended the formalism to nonextensive statistics. ► Adding memory terms leads to the formation of positive kappa-like distributions.► Negative kappa-like exponents are related to enhanced interaction of increased order in the system.► Stronger memory terms result in kappa-like distributions with lower kappa.