Random-Walk Interpretation and Generalization of Linear Boltzmann Equations, Particularly for Neutron Transport

The connection between linear recurrence relations which define generalized random walks and the related linear Boltzmann equations is clarified. The probability distribution $f_n\mathrm{}\left(s\right)\mathrm{}$ for "the state $s$" reached by a "random walker" after $n$ steps satisfies the recurrence relation $f_{n+1\mathrm{}}\mathrm{}\left(t\right)=\int f_n\mathrm{}\left(s\right)P(s, t)ds$, where the non-negative $P(s, t)$ is the probability for a transition from $s$ to $t$. The Boltzmann distribution is given by $f\left(s\right)={{\Sigma }_{n=0\mathrm{}}}^{\infty }{f}_n\mathrm{}\left(s\right)\mathrm{}$. In general, $f_n\mathrm{}\left(s\right)\mathrm{}$ contains more information that $f\left(s\right)\mathrm{}$. Moreover, $f_n\mathrm{}\left(s\right)\mathrm{}$ is the $n\mathrm{th}$ term in the iteration series solution of the Boltzmann equation and therefore can also be obtained from the solution of an associated Boltzmann equation which contains an additional parameter. As an example, the well-known integral Boltzmann equation for neutron transport in a nonmultiplying infinite medium is derived from a $P(s, t)$ which involves a transition in a seven-dimensional phase-time space. Brownian motion and Rayleigh's problem (related to neutron thermalization) may be treated similarly.