Binomial vs. Poisson statistics: From a toy model to a stochastic model for radioactive decay

Disintegration is a physical phenomenon of atomic nuclei —radioactive isotopes decay— has been modeled with different approaches (deterministic and random), from didactic toy models using as reference the roll (experiment) of standard six-sided dice (Arthur and Ian, 2012), to the generalization of probabilistic methods. Using for example the moment generating function (MGF) method, to obtain the behavior of its probability distribution for n multi-sided dice, —i.e., dice of s≥6 sided (Singh et al., 2011, Sánchez-Sánchez et al., 2022)—. Radioactive decay is essentially statistical (random) in nature, so we cannot predict when any of the atoms will decay. The MGF method and stochastic models were applied to the so-called radioactive dice (toy model), to obtain a theoretical Poisson-like distribution (stochastic model). In this work, we carry out an exhaustive study and a comparison —on the apparent discrepancy— of the Binomial and Poisson statistics associated with the decay of radioactive nuclei. To gain a deeper understanding of this phenomenon —radioactive decay— we use the theory of stochastic processes, i.e., modeling these distributions in a stochastic context using the efficient mathematical tools of this theory. They are discrete random variable processes in continuous time. So, we use an approach of the so-called master equation (Kolmogorov equations). We study them in general as Birth-Death processes —both processes separately, highlighting their disagreements— modeling the Poisson process as a Pure Death process. We solve the master equations of the Poisson process by introducing the so-called sojourn time. Also, we study the relative fluctuations through the Fano factor. We analyze the deeper concept of Entropy of the binomial and Poisson processes by calculating their metrics —In Shannon’s information theory context—. We show that they have a corresponding statistical link between both images and the radioactive decay distributions. In this way, we gain a deeper insight into the random nature of nuclear decay with its stochastic distributions. In summary, this paper addresses the extension of deterministic systems that are, in reality, of a random nature within a theoretical framework of so-called stochastic processes and information theory (entropy). We link entropy stochastic (binomial and Poisson) and its intrinsic fluctuations with the physical mechanisms of the collective dynamics of radioactive decay.