Abstract
The eigenvalue problem in Theorem 4.1 is not the only one that offers insight into the evolution of neutron transport. In this chapter we will consider two different time-independent, or stationary, eigenvalue problems. The one that is of most interest to the nuclear industry is known as the \(k_{\mathtt {eff}}\)-eigenvalue problem. Roughly speaking, the eigenvalue \(k_{\mathtt {eff}}\) has the physical interpretation as being the ratio of neutrons produced during fission events to the number lost due to absorption, either at the boundary or in the reactor due to neutron capture. As such, it characterises a different type of growth to the eigenvalue problem considered in the continuous-time setting in Theorem 4.1. In this chapter, in a similar fashion to the time-dependent setting, we will explore the probabilistic interpretation of \(k_{\mathrm {eff}}\) and its relation to the classical abstract Cauchy formulation.
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Notes
- 1.
Here, we define \(\prod _{i = 1}^0\cdot := 1\).
- 2.
To be precise, by a positive eigenfunction, we mean a mapping from \(D\times V\to (0,\infty )\). This does not prevent it being valued zero on \(\partial D\), as D is open.
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Horton, E., Kyprianou, A.E. (2023). Generational Evolution. In: Stochastic Neutron Transport . Probability and Its Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-39546-8_7
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