Joint Moments of Neutron Number and Medium State in a Time-varying Multiplying Medium

The zero-power reactor noise theory in a steady neutron multiplying medium was extended recently to a medium randomly varying in time. However, it was found that, for a time-varying medium where the transition probability randomly fluctuates, the treatment of the joint moments of the neutron number N and the medium state η by the forward master equation approach leads to a closure problem. The present paper describes and demonstrates a moment calculation technique that can solve this closure problem exactly in a time-varying binary multiplying medium where η has two possible realizations. In addition to the first and the second moments of N alone, the asymptotic values of the joint moments of N and η in the time-varying multiplying medium were also obtained in a compact form for n = 1, 2 and an arbitrary value of m, without closure assumptions. It was found that, for even m values, the asymptotic values of the moments of N and η are uncorrelated, whereas for odd m values, they are negatively correlated, i.e., their covariance is less than zero. Furthermore, a perfect agreement was found between the Monte Carlo solutions and the theoretical expressions derived.