This PhD proposal aims to develop a fusion-specific T-effective model based on stochastic probability theory, with particular emphasis on the neutron transport equation and the stochastic nature of tritium production, transport, and fuelling. Tritium, which is bred in a breeder blanket, must be extracted, purified, and re-injected into the plasma to sustain the fusion reaction. These processes are inherently probabilistic and could be modelled stochastically. We propose to develop probabilistic models for each stage of the neutron-tritium life cycle which can be included into a generalization of the neutron-Boltzmann equation. This equation could then be re-written to produce a definition of T-effective, where a T-effective < 1 means a reactor will not work, T-effective = 1 means a sustainable reactor and T-effective > 1 means surplus tritium production. This builds on the classical notion of tritium breeder ratio (TBR), re-interpretating it probabilistically and includes the tritium recovery, processing and subsequent reaction. T-effective, like K-effective, can be interpreted as the probability that a tritium in the plasma will produce a neutron which will eventually produce a tritium (after production, extraction etc) which will react with a deuterium. This is intended to be the start of a longer-term program, which will evolve (the probabilistic models of each stage will be refined) as new science is developed.

Research Objectives

1. Mathematical Definition of Fusion-Specific T-Effective: Develop a stochastic version of k-effective that accurately represents the complex dynamics of fusion reactors. Producing a probability that a D-T neutron will ultimately produce a second D-T neutron, considering a model for tritium production, extraction etc. A renewal theory type approach will be used to model the probabilistic lifetime of neutrons in the breeder blanket, linking to a fusion power plants fuel performance.

2. Stochastic Tritium Transport: Develop a stochastic tritium transport framework. This will be slower than the current deterministic approach, but it will be much simpler to set up for a complex geometry. This is analogous to why Monte Carlo neutron transport is used in fusion as opposed to deterministic (Sn) neutron transport. This will be part of the basis for the statistical analysis and an entire new basis to transport tritium.

3. Coupling of Neutron and Tritium Transport: Develop coupled stochastic models for neutron transport and tritium production. This will allow for a comprehensive analysis of how neutron flux influences tritium production, and how uncertainties in tritium transport affect overall reactor performance. Coupled SPDEs (stochastic partial differential equations) will be used to model the interdependent processes of neutron transport and tritium production. Solving these equations simultaneously, we can assess how variations in neutron transport propagate through the system to influence tritium production, extraction, and re-injection, providing insights into reactor stability and efficiency under varying conditions. The coupling will then allow for a holistic probabilistic framework to be developed, leading to a version of T-effective to be formulated.