Asymptotic Behaviour of Neutron Transport Processes

The solution of the initial-value problem of the time-dependent linear Boltzmann equation corresponds to a semigroup of linear transformations: Initial-value problem: δn/δt = An n(x, v, 0) = f(x, v) Solution: n(x, v, t) = Ttf(x, v) Tt = eAt Lehner and Wing were the first to use a Laplace transform technique for the study of the asymptotic behaviour of the solution. Mika, Bednarz, Albertoni, Kaper et al. extended this technique to more general problems. The main task is always to find the spectrum of the Boltzmann operator A. Asymptotic behaviour is closely related to the point spectrum of A , if the latter exists. This paper uses a completely new approach, mean ergodic theory, the study of asymptotic properties of semigroups of bounded transformations in a Banach space. Two types of function spaces are considered; (1) The Banach space Ltm of functions integrable on the six-dimensional μ- space of statistical mechanics. The Banach norm ||f|| equals the total number of neutrons in the system; (2) The Hilbert space L2m of functions square integrable on the μ-space. The inner product (f, g) equals the totalNcount rate of the neutron distribution f due to an array of neutron detectors described by the weight function g of the space dual to that of all neutron distributions. Excluding the case of a super-critical system, the semigroup generated by the Boltzmann operator A is uniformly bounded || Tt II K, t ≥ 0. The splitting theorem of mean ergodic theory can be applied to T t . The initial distribution is split uniquely into a sum of a reversible and a flight vector. Now a special property of the semigroup generated by the Boltzmann operator A enters: there exists a characteristic time t0 > 0 depending only on the geometry and chemistry of the system such that Ttf(x,v) > 0 for all (x, v) , for all f and all t ≥ 0. From this property it is possible to deduce that an equilibrium distribution exists and is unique. Taking the Hilbert space L2m criticality of a reactor corresponds to strong mixing in the sense of ergodic theory; we define a reactor as critical if for all f and all g, positive almost everywhere, a positive limit (Ttf, g) exists for t --> ∞. This definition corresponds to the Fermi experiment. Boundedness of Tt can be demonstrated. Finally an attempt is made to define the mean entropy of a neutron transport process. (author)