Nuclear Physics Division

Branching Brownian motion and phase transition

een from a distance, the random walk of neutrons follows a Brownian motion, in that it obeys the diffusion equation. The characteristic trajectories are erratic and noisy (see Figure 1.a): mathematicians describe them as continuous everywhere, but derivable nowhere. Nuclear power reactors are operated using this equation (or, to be more precise, using the diffusion equation with two groups of energies: one equation to describe the transport of fast neutrons, and another for slower, so-called thermal neutrons). In this way, the behavior of the neutron population in a reactor can be accurately modeled.

However, when the power in the reactor is stationary, the study of the chain reaction proves more delicate. Neutron scattering is termed "critical" because, under these conditions, a neutron gives rise, on average, to precisely one other neutron during its lifetime, but each interaction of a neutron with a heavy nucleus can potentially give rise to several "son" neutrons, resulting in branching trajectories. The study of branching Brownian motion (BBM for short), shown in Figure 1.b, was used some years ago to understand the onset phase of epidemics [1], and similar models were then used in neutronics to characterize the statistical fluctuations of the neutron gas. These can develop to the point where the gas no longer follows the diffusion law: this is known as a phase transition. In this "phase", the total neutron population can, in theory, become extinct or much larger than expected, leading ipso facto to a non-negligible increase in reactor power. In theory only, because in practice, recent research has shown that the total number of neutrons in the reactor fluctuates very little [2,3]. This surprising stability is due to the temperature effect: the higher the power (i.e. the number of neutrons) locally, the higher the temperature, and the more neutrons tend to be captured by the medium rather than inducing fission. This counter-reaction is due to the broadening of neutron resonances, which literally become neutron traps.

The results of their work, published in the journal Physical Review E [4], brought to light some relatively surprising conclusions: the critical diffusion equation characterizing the neutron population becomes, in the presence of counter-reactions, a so-called super-diffusion equation, where the diffusive term usually given by a spatial derivative of order 2 in space then takes on a non-integer value, slightly less than 2! The microscopic generator of this equation (which describes the laws of neutron transport) is no longer a Brownian motion, but a Lévy random walk: the trajectories appear very similar to those of Brownian motion, with the exception of certain "jumps" projecting the neutrons a long way from their initial position (see Figure 1.c). As a result, the spatial distribution of power in the reactor is slightly modified (by a few percent), as shown in Figure 2, where the super-diffusion result is dotted and compared with that of simple diffusion. This is about the same order of magnitude as the accuracy expected from the calculation codes used in nuclear safety studies... So it seems that the safety ball is now in the numerical engineers' court!