A Diffusion Scale Approximation for Stochastic Fission Chains with State Feedback

Modeling and simulation of stochastic fission chains, often referred to as zero power reactor noise or stochastic transport, is a central topic in nuclear science and engineering, with important practical applications in reactor control and measurements. The common mathematical setting for studying reactor noise is through branching processes, typically modeled using forward and backward master equations. Because of the high complexity of the problem, reactor noise is typically studied under the point model approximation and in a linear setting, which neglects any reactivity feedbacks. On the other hand, since reactivity feedbacks are dominant in power reactors, stochastic models and simulations in a nonlinear setting are of great interest. For this reason, there is a constant interest in new mathematical frameworks that will enable modeling and simulation in a complete fashion. In the present study, we look at a branching process with a negative feedback, realized by a linear increase in the per capita death rate. In particular, the outline of the study is to compare two models for the above process. The first is the exact model, based on a forward master equation, and the second is the diffusion scale approximation (or the functional central limit approximation), realized in an Ito-type stochastic differential equation (SDE). The comparison between the two models is done in two levels. First, we show that the dynamics of the first and second moments of both models are governed by the same equations. Then, by realization of each model, we contract a Monte Carlo simulation for sampling the population size distribution. In particular, we show that once the population size is sufficiently large, the SDE approximation allows us to perform accurate simulations in a critical setting, with a dramatic reduction of the CPU time.