Stochastic Equation Approach to Power Spectrum Modeling of Fuel Debris

Modeling of the power spectrum (PS) is an essential issue in criticality analysis of continuously mixed random media. Image analysis of a fuel debris mock-up reveals an inverse power law PS with a flattening tendency in the low spectral domain and an accelerating decrease in the mid to high spectral domain. In this paper, these characteristic features are reproduced via stochastic differential equations (SDEs) as follows. First, by constraining a dominant order of magnitude for the spectrum domain variable, the inverse power law is derived using the variational principle under the condition of being as disordered as possible. Second, to show how deviations from a pure inverse power law arise from realistic physical effects, a second-order ordinary differential equation with white noise is formulated by taking frictional and restoring effects separately into account. This equation is then reduced to a system of first-order SDEs driven by Brownian motion within the framework of Itô stochastic calculus. Third, based on an integrator derived from the operator splitting of the SDEs, numerical results for PS are presented to demonstrate the flattening tendency in the low spectral domain and the steeper-than-inverse-square decay in the high spectral domain. It is also shown that PS approaches the inverse square law in the limit of large friction so that the effect of restoration is negligible. Finally, SDEs are proposed to generate PSs for use as input to a randomization function for the volume fractions of constituent materials in random media criticality calculations.