On the Distributions of Absorbed Particles in Crossing a Field Containing Absorption Points

Abstract

Consider a queue of particles that are required to cross a field containing a random number of absorption points (traps) acting independently. Suppose that if a particle clashes with (contacts) any of the absorption points, it is absorbed (trapped) with probability p and non absorbed with probability q = 1 − p. Let X_n be the number of absorbed particles from a queue of n particles and T_k the number of particles required to cross the field until the absorption of k particles. Assuming that the number Y of absorption points in the field obeys a q-Poisson distribution (Heine or Euler distribution), the distributions of X_n and T_k are obtained as q-binomial and q-Pascal distributions, respectively. Inversely, assuming that X_n obeys a q-binomial distribution (or, equivalently, assuming that T_k obeys a q-Pascal distribution), the distribution of Y is obtained as a q-Poisson distribution (Heine or Euler distribution).