Stochastic Methods for Neutron Transport Equation III: Generational many-to-one and $k_{\texttt{eff}}$

Following e.g. [6, 7, 15], we want to formulate (1.1) in the function space $L_{2}(D\times V)$. The so-called (initial-value) abstract Cauchy problem (ACP) takes the form

where $\mathcal{A}={\mathcal{T}}+{\mathcal{S}}+{\mathcal{F}}$ and ${{u}}_{t}$ belongs to the space $L_{2}({D}\times V)$, for $t\geq 0$ (in particular $g\in L_{2}({D}\times V)$). Specifically, $({{u}}_{t},t\geq 0)$ is continuously differentiable in the space $L_{2}({D}\times V)$, meaning there exists a $\dot{{u}}_{t}\in L_{2}({D}\times V)$, which is time-continuous in $L_{2}({D}\times V)$ with respect to $\lVert\cdot\rVert_{2}$ and such that $\lim_{h\to 0}h^{-1}({u}_{t+h}-{u}_{t})=\dot{{u}}_{t}$ for all $t\geq 0$. Necessarily, the solution to (2.1) forms a $c_{0}$-semigroup³³3Recall that a $c_{0}$-semigroup $(\texttt{V}_{t},t\geq 0)$ also goes by the name of a strongly continuous semigroup and, in the present context, this means it has has the properties that (i) $\texttt{V}_{0}={\rm Id}$, (ii) $\texttt{V}_{t+s}[g]=\texttt{V}_{t}[\texttt{V}_{s}[g]]$, for all $s,t\geq 0$, $g\in L_{2}({D}\times V)$ and (iii) for all $g\in L_{2}({D}\times V)$, $\lim_{h\to 0}\lVert\texttt{V}_{h}[g]-g\rVert_{2}=0$.. Moreover, ${\rm Dom}(\mathcal{A}):=\{g\in L_{2}(D\times V):\upsilon\cdot\nabla g\in L_{2}(D\times V)\text{ and }g|_{\partial D^{+}}=0\}$ is the domain of $\mathcal{A}$ and ${{u}}_{t}\in{\rm Dom}(\mathcal{A})$ for all $t\geq 0$.

In the ACP setting, the notion of an eigenpair $(\lambda,\varphi)$ is well formulated on $L_{2}(D\times V)$ via (1.4). Equivalently, it means we are looking for $\varphi\in L^{+}_{2}(D\times V)$ and $\lambda$ such that $\texttt{V}_{t}[\varphi]={\rm e}^{\lambda t}\varphi$ on $L^{+}_{2}(D\times V)$, for all $t\geq 0$. The sense in which we mean that $\lambda$ is a ‘leading’ eigenvalue roughly boils down it corresponding to the eigenvalue in the spectrum of the operator $\mathcal{A}$ on $L_{2}(D\times V)$ with the largest real part (and, as usual, it is real valued itself), and moreover, its associated eigenfunction $\varphi$ is non-negative. As such, one expects the existence of a non-negative left eigenfunction $\tilde{\varphi}$ (e.g. in the sense that $\langle\tilde{\varphi},\texttt{V}_{t}[g]\rangle={\rm e}^{\lambda t}\langle\tilde{\varphi},g\rangle$ for $t\geq 0$) such that

as $t\to\infty$. Here, we use the notation $\langle f,g\rangle=\int_{D\times V}f(r,\upsilon)g(r,\upsilon){\textnormal{d}}r{\textnormal{d}}\upsilon$, so that $\lVert\cdot\rVert_{2}=\langle\cdot,\cdot\rangle^{1/2}$. Precise results of this nature can be found in [7, 15, 5].

The evolution of $(X_{t},t\geq 0)$ is a stochastic process valued in the space of atomic measures $\mathcal{M}(D\times V):=\{\textstyle{\sum_{i=1}^{n}}\delta_{(r_{i},\upsilon_{i})}:n\in\mathbb{N},(r_{i},\upsilon_{i})\in D\times V,i=1,\cdots,n\}$ which evolves randomly as follows.

A particle positioned at $r$ with velocity $\upsilon$ will continue to move along the trajectory $r+\upsilon t$, until one of the following things happens.

1. (i)

   The particle leaves the physical domain $D$, in which case it is instantaneously killed.

2. (ii)

   Independently of all other neutrons, a scattering event occurs when a neutron comes in close proximity to an atomic nucleus and, accordingly, makes an instantaneous change of velocity. For a neutron in the system with position and velocity $(r,\upsilon)$, if we write $T_{\texttt{s}}$ for the random time that scattering may occur, then independently of any other physical event that may affect the neutron, $\Pr(T_{\texttt{s}}>t)=\exp\{-\textstyle{\int_{0}^{t}}\sigma_{\texttt{s}}(r+\upsilon s,\upsilon){\rm d}s\},$ for $t\geq 0.$

   When scattering occurs at space-velocity $(r,\upsilon)$, the new velocity $\upsilon^{\prime}\in V$ is selected independently with probability $\pi_{\texttt{s}}(r,\upsilon,\upsilon^{\prime}){\textnormal{d}}\upsilon^{\prime}$.

3. (iii)

   Independently of all other neutrons, a fission event occurs when a neutron smashes into an atomic nucleus. For a neutron in the system with initial position and velocity $(r,\upsilon)$, if we write $T_{\texttt{f}}$ for the random time that scattering may occur, then, independently of any other physical event that may affect the neutron, $\Pr(T_{\texttt{f}}>t)=\exp\{-\textstyle{\int_{0}^{t}}\sigma_{\texttt{f}}(r+\upsilon s,\upsilon){\rm d}s\},$ for $t\geq 0.$

   When fission occurs, the smashing of the atomic nucleus produces lower mass isotopes and releases a random number of neutrons, say $N\geq 0$, which are ejected from the point of impact with randomly distributed, and possibly correlated, velocities, say $\{\upsilon_{i}:i=1,\cdots,N\}$. The outgoing velocities are described by the atomic random measure

   $$\mathcal{Z}(A):=\sum_{i=1}^{N}\delta_{\upsilon_{i}}(A),\qquad A\in\mathcal{B}(V).$$

   (2.4)

   If such an event occurs at location $r\in\mathbb{R}^{d}$ from a particle with incoming velocity $\upsilon\in{V}$, we denote by ${\mathcal{P}}_{(r,\upsilon)}$ the law of $\mathcal{Z}$. The probabilities ${\mathcal{P}}_{(r,\upsilon)}$ are such that, for $\upsilon^{\prime}\in{V}$, for bounded and measurable $g:V\to[0,\infty)$,

   $\displaystyle\int_{V}g(\upsilon^{\prime})\pi_{\texttt{f}}(r,v,\upsilon^{\prime}){\textnormal{d}}\upsilon^{\prime}$

   $\displaystyle={\mathcal{E}}_{(r,\upsilon)}\left[\int_{V}g(\upsilon^{\prime})\mathcal{Z}({\textnormal{d}}\upsilon^{\prime})\right]=:{\mathcal{E}}_{(r,\upsilon)}[\langle g,\mathcal{Z}\rangle].$

   (2.5)

   Note, the possibility that $\Pr(N=0)>0$, which will be tantamount to neutron capture (that is, where a neutron slams into a nucleus but no fission results and the neutron is absorbed into the nucleus).

The NBP is thus parameterised by the quantities $\sigma_{\texttt{s}},\pi_{\texttt{s}},\sigma_{\texttt{f}}$ and the family of measures ${\mathcal{P}}=({\mathcal{P}}_{(r,\upsilon)},r\in D,\upsilon\in V)$ and accordingly we refer to it as a $(\sigma_{\texttt{s}},\pi_{\texttt{s}},\sigma_{\texttt{f}},\mathcal{P})$-NBP. It is associated to the NTE via the relation (2.5), and, although a $(\sigma_{\texttt{s}},\pi_{\texttt{s}},\sigma_{\texttt{f}},\mathcal{P})$-NBP is uniquely defined, a NBP specified by $(\sigma_{\texttt{s}},\pi_{\texttt{s}},\sigma_{\texttt{f}},\pi_{\texttt{f}})$ alone is not. Nonetheless, it is easy to show that for a given $\pi_{\texttt{f}}$, a $(\sigma_{\texttt{s}},\pi_{\texttt{s}},\sigma_{\texttt{f}},\mathcal{P})$-NBP always exists which satisfies (2.5). See the discussion in Section 2 of [10].

Define

where $\mathbb{P}_{\delta_{(r,\upsilon)}}$ is the law of $X$ initiated from a single particle with configuration $(r,\upsilon)$, and $g\in L^{+}_{\infty}(D\times V)$, the space of non-negative uniformly bounded measurable functions on $D\times V$. Here we have made a slight abuse of notation (see $\langle\cdot,\cdot\rangle$ as it appears in (2.5)) and written $\langle g,X_{t}\rangle$ to mean $\textstyle{\int_{D\times V}}g(r,\upsilon)X_{t}({\textnormal{d}}r,{\textnormal{d}}\upsilon)$. The following result was shown in [5, 10, 7, 6].

In [5] the below result was shown, which demonstrates the context in which the mild solution to the NTE and the ACP can be seen to coincide.

In the probabilistic setting, the meaning of (1.4) can be interpreted as looking for a pair $\lambda$ and $\varphi$ such that, pointwise on $D\times V$, $\psi_{t}[\varphi]={\rm e}^{\lambda t}\varphi$, for $t\geq 0$. As alluded to in (1.3), we have a similar asymptotic to (2.2), which isolates the eigenpair $(\lambda,\varphi)$ in its limit. The notion of ‘leading’ in the probabilistic setting is less obvious, however, due to Theorem 2.3, the eigenpairs that emerge from the mild setting and the weak linear PIDE setting should in principle agree on $L_{2}(D\times V)$. This is discussed with greater precision in [5, 10].

There is a second stochastic representation of the unique solution to (2.7), which makes use of the so-called neutron random walk (NRW). This process takes values in $D\times V$ and is defined by its scatter rates, $\alpha(r,\upsilon)$, $r\in D,\upsilon\in V$, and scatter probability densities $\pi(r,\upsilon,\upsilon^{\prime})$, $r\in D,\upsilon,\upsilon^{\prime}\in V$. When issued with a velocity $\upsilon$, the NRW will propagate linearly with that velocity until either it exits the domain $D$, in which case it is killed, or at the random time $T_{\texttt{s}}$ a scattering occurs, where $\Pr(T_{\texttt{s}}>t)=\exp\{-\textstyle{\int_{0}^{t}}\alpha(r+\upsilon t,\upsilon){\rm d}s\},$ for $t\geq 0.$ When the scattering event occurs at position-velocity configuration $(r,\upsilon)$, a new velocity $\upsilon^{\prime}$ is selected with probability $\pi(r,\upsilon,\upsilon^{\prime}){\textnormal{d}}\upsilon^{\prime}$. If we denote by $(R,\Upsilon)=((R_{t},\Upsilon_{t}),t\geq 0)$, the position-velocity of the resulting continuous-time random walk on $D\times V$ with an additional cemetery state for when it leaves the domain $D$, it is easy to show that $(R,\Upsilon)$ is a Markov process. We call the process $(R,\Upsilon)$ an $\alpha\pi$-NRW.

In order to solve the $k$-eigenvalue problem (1.6), it turns out that $(\psi_{t},t\geq 0)$ and $(\phi_{t},t\geq 0)$ are not the right objects to work with on account of their time-dependency. We now consider a generational model of the NBP. We can think of each line of descent in the sequence of neutron creation as genealogies. In place of $(X_{t},t\geq 0)$, we consider the process $(\mathcal{X}_{n},n\geq 0)$, where, for $n\geq 1$, $\mathcal{X}_{n}$ is $\mathcal{M}(D\times V)$-valued and can be written $\mathcal{X}_{n}=\sum_{i=1}^{\mathcal{N}_{n}}\delta_{(r^{(n)}_{i},\upsilon^{(n)}_{i})}$, where $\{(r^{(n)}_{i},\upsilon^{(n)}_{i}),i=1,\cdots\mathcal{N}_{n}\}$ are the position-velocity configurations of the $\mathcal{N}_{n}$ particles that are $n$-th in their genealogies to be the result of a fission event. $\mathcal{X}_{0}$ is consistent with $X_{0}$ and is the initial configuration of neutron positions and velocities. As such, for $n\geq 1$ we can think of $\mathcal{X}_{n}$ as the $n$-th generation of the system and we refer to them as the neutron generational process (NGP). The reader who is more experienced with the theory of branching processes will know $\mathcal{X}_{n}$ to be an example of what is called a stopping line; see [12].

Appealing to the obvious meaning of $\langle g,\mathcal{X}_{n}\rangle$, define the expectation semigroup $(\Psi_{n},n\geq 0)$ by

with $\Psi_{0}[g]:=g\in L_{\infty}^{+}(D\times V)$. The main motivation for introducing the NGP is that, just as we have seen that the meaning of (1.4) can be phrased in terms of a multiplicative invariance with respect to the solution of an ACP (2.1) or of the mild equation (2.7), we want to identify the eigen-problem (1.6) in terms of the semigroup above.

To this end, let us introduce the problem of finding a pair $k>0$ and $\phi\in L^{+}_{\infty}(D\times V)$ such that, pointwise,

In the next section we will show the existence of a solution to (2.12) which also plays an important role in the asymptotic behaviour of $\Psi_{n}$ as $n\to\infty$. Before getting there, let us give a heuristic argument as to why (2.12) is another form of the eigenvalue problem (1.6).

By splitting on the first fission event, $\Psi_{n}$ solves the following mild equation

where $(\texttt{Q}_{s},s\geq 0)$ is the expectation semigroup associated with the operator $\mathcal{T}+\mathcal{S}$. More precisely,

where $(R_{s},\Upsilon_{s})_{s\geq 0}$ is the $\sigma_{\texttt{s}}\pi_{\texttt{s}}$-NRW. Then, if the pair $(k,\phi)$ solves (2.12), the strong Markov property along with an iteration implies that

Using it in (2.13) and dividing through by $k^{n}$ yields

Now set

Then, heuristically speaking, since Q is associated to the generator $\mathcal{T}+\mathcal{S}$, classical Feynman-Kac theory suggests that $V_{t}$ ‘solves’ the equation

with $V_{0}=0$. Note that $\partial V_{t}/\partial t=\texttt{Q}_{t}[g]$, which tends to zero as $t\to\infty$ thanks to the transience of $(R,\Upsilon)$. Hence, taking $g={k}^{-1}\mathcal{F}\phi$, letting $t\to\infty$ in the above equation, recalling that $(\texttt{Q}_{s},s\geq 0)$ is the expectation semigroup associated with the operator $\mathcal{T}+\mathcal{S}$, and using the identity (2.15) yields