Stochastic Point Kinetics Model of Circulating-Fuel Reactors under Perfect Mixing Approximation

We present a stochastic framework for low-population dynamics in circulating-fuel reactors (CFRs) that captures delayed-neutron precursor (DNP) transport without delay terms. Starting from a modified point-kinetics model with two perfectly-mixed volumes, we derive equivalent discrete-event dynamics and an Itô stochastic differential equation (SDE) system. Two solvers are implemented: an analog Monte Carlo (AMC) engine and a semi-implicit Milstein SDE solver. Transient benchmarks demonstrate perfect agreement of AMC/SDE means with deterministic solutions, while revealing that the SDE approach underestimates DNP variances in selected regimes, potentially due to the neglect of DNP noise. We further recast reactivity loss due to precursor drift in this stochastic setting and show that its estimator is negatively biased. Overall, the developed framework provides a minimal yet representative model for CFR low-population kinetics. Future work will re-derive and test SDE noise terms and apply the framework to selected transient applications such as start-up analyses of CFRs.

💡 Research Summary

The paper introduces a stochastic framework for low‑population kinetics in circulating‑fuel reactors (CFRs), focusing on the transport of delayed‑neutron precursors (DNPs) without explicit delay terms. By representing the reactor as two perfectly mixed control volumes—the core (c) and the ex‑core (e)—the authors replace the traditional delay term in the point‑kinetics equations with residence‑time based advection‑reaction terms. This yields a set of three ordinary differential equations for the neutron population N(t) and the precursor inventories C_c(t) and C_e(t), which are Markovian and thus amenable to stochastic treatment.

The stochastic description proceeds in two parallel ways. First, an event‑based analog Monte‑Carlo (AMC) solver called MARS directly simulates the discrete events listed in Table 1 (neutron loss, prompt and delayed fission, external source, precursor decay, and inter‑volume transfer). Each event follows a Poisson process with rates derived from the underlying point‑kinetics parameters β, Λ, ρ(t), and the residence times τ_c, τ_e. Second, the authors derive an Itô stochastic differential equation (SDE) system by taking the diffusion limit of the discrete process. The SDE for N(t) contains a drift term identical to the deterministic point‑kinetics equation and a diffusion term D(t) |N(t)| dW_t, where D(t) is a function of the variance of the prompt‑neutron multiplicity and the loss rate γ(t). The precursor equations contain only deterministic drifts; stochastic diffusion terms are omitted.

Both solvers are implemented and benchmarked against a “ramp‑up” transient that simultaneously varies reactivity and residence times. Parameters are chosen to satisfy the source‑positivity condition required for the SDE to remain almost surely positive. The AMC simulations use up to 10 000 independent histories, while the SDE solver uses 8 000 Milstein trajectories with a semi‑implicit time‑stepping scheme. Results show perfect agreement of the mean trajectories from both stochastic methods with the deterministic solution, confirming that the stochastic framework reproduces the expected average dynamics.

However, variance comparison reveals a systematic under‑estimation by the SDE approach for certain precursor groups, especially the in‑core group 1. The authors attribute this discrepancy to the omission of stochastic terms in the precursor equations; the DNP noise is effectively set to zero in the diffusion limit they derived. This leads to narrower confidence bands for the SDE results compared with the AMC reference, indicating that the current SDE formulation does not capture the full stochasticity of the precursor population.

The paper also investigates the estimation of reactivity loss due to precursor drift, a quantity of interest for safe start‑up analyses. Using the derived expression ρ₀ = β − (Λ N₀)⁻¹ ∑ λ_j C_{j,c,0}, the authors compute an estimator ˆρ₀ from stochastic simulations. Because ρ₀ is a nonlinear function of the ratio C_c/N, Jensen’s inequality predicts a negative bias in E