Boltzmann Transport Equation

The starting point of many radiation transport models is the Boltzmann transport equation, which has a rather complicated integro-differential structure such as the one above. The coefficients appearing in this equation all pertain to particle collision, fission, absorption, scatter, energy depletion and radiation source and are highly inhomogeneous through their dependency on the three variables (r,Ω,E). Finding exact solutions to such equations is out of the question and the use of Feynman-Kac representation (averages over paths of stochastic processes) plays an important role in informing Monte-Carlo simulation methods.

Neutron transport

A simple case in point is the setting of understanding criticality of nuclear reactors. The complexity of the domain over which the Boltzmann transport equation is determined by the geometry of the reactor.

Understanding whether a reactor design can be held in a critical or close to critical state (the normal operational mode for nuclear reactors) is entirely determined by the leading eigen solution to the Boltzmann transport equation.

In this setting the Feynmann-Kac representation equation is not unique and one can appeal to a representation involving either spatial branching process or a kind of continuous random walk. In this project, IMI affiliated researchers and their national and international collaborators have worked with industrial partner, the ANSWERS group at Jacobs, looking at different stochastic representations of the time-evolving neutron transport equation and how this bears relevance to possible Monte Carlo algorithms and their complexity.