Here, all individuals have the same mortality (death) rate ц, and the transition rates are
State 0 is an absorbing state, whereas other states are transient, which is shown in Figure 10.2.

Pure Birth Process

Here, the birth probability per unit time is X, and the transition rates are
Initially, the population size is 0 and all the transition rates are shown in Figure 10.3.

FIGURE 10.2

Transition of states in the pure death process.

FIGURE 10.3

Transition of states in the pure birth process.

Stochastic Point Kinetic Model

The point kinetic equation has been modelled in terms of stochastic (Hayes and Allen 2005) by considering the birth and death processes of the neutron and the precursor population. The coupled deterministic time-dependent equations for the neutron density and the delayed neutron precursors may be represented as

for i = 1,2 , ... , m where

N = N(r, t) is the neutron density at position r at time t

v is the velocity of the neutron

Xf is the neutron fission cross section

D is the diffusion coefficient

X_{a} is the absorption coefficient

Zrn

bi is the delayed neutron fraction

1 - p is the prompt neutron fraction k_{x} is the infinite medium neutron reproduction factor is the delay constant, Q=C_{i}(r, t) density of the ith type of precursors at position r at time t S_{0} is the extraneous neutron source DvV^{2}N is the diffusion term of the neutrons (X_{a} - Xf) is the capture cross section

[(1 - b)k^Xa - EfJvN is the prompt neutron contribution to the source

Zm

k_{t}C_{t} is the rate of transformations from the neutron precursors to the neutron population

In the present investigation, the neutron captures and the fission process are considered as deaths and pure birth process, respectively. Here, v(1 - p) - 1 neutrons are born in each fission along with the precursor fraction vp.

To apply the separation of variables technique, let us consider N =f(r)n(t) and Q=g(r)c(t), where we assume that N and Q are separable in time and space. Now, Equation 10.2 becomes

It is assumed here that f/g is independent of time and (f(r)/g(r)) = 1.

As such, we have

By making the same substitutions as made earlier, Equation 10.1 becomes

We assume that f satisfies V^{2}/ + B^{2}f = 0 (a Helmholtz equation) and that S_{0} has the same spatial dependence as f. Thus, we will have q(t) = (S_{0}(r, t)/f(r)). Equation 10.5 that describes the rate of change of neutrons with time is

These equations represent a population process where n(t) is the population of neutrons and c(t) is the population of the ith precursor. We separate the neutron reactions into two terms: deaths and births. Therefore, we have Equations 10.6 and 10.4 as

This system is now to be solved by introducing the following symbols:

The absorption lifetime and the diffusion length are represented as l_{m} = 1/vL_{a} and L^{2}=D/L_{a}. Equation 10.7 becomes

After simplification and regrouping, Equation 10.9 becomes
Performing the same substitutions in Equation 10.8 gives

Again, two more constants k=k^/(1 + L^{2}B^{2}) and l_{0} = l^/(1 + L^{2}B^{2}) are introduced, viz. the reproduction factor and the neutron lifetime, respectively. Now Equation 10.10 becomes

After the substitution and rearranging of Equation 10.12, we have
Similarly, Equation 10.11 will be

Next, we consider these equations in terms of the neutron generation time. We define l = l_{0}/k as the generation time and then Equations 10.13 and 10.14 become

Next, the reactivity is defined as p = 1 - (1/k) and Equation 10.15 becomes
For further simplification, we consider the term l_{If}/(L_{a}l_{x>}), which may be written as
where a is defined as a = 5f/(?_{a}k_{OT}). Equation 10.17 will be
Finally, this deterministic system may be written as

for i = 1,2 , ... , m.

Here, we have considered the population size of neutrons as n and the population size of the ith neutron precursor as c,. The neutron birth rate due to fission is b = (1 - a - p)/ (l(-1 + (1 - p)v)), where (-1 + (1 - p)v) is the number of new neutrons born in each fission, and d = (-p + 1 - a)/l is the neutron death rate due to captures and leakage, whereas is the rate at which the ith precursor is transformed into neutrons and q is the rate at which the source neutrons are produced.

For simplicity, consider one precursor to derive the stochastic system for the population dynamics. Note that for one precursor p = p_{1}, p is used for one precursor to represent the total delayed neutron fraction. This notation will be used for generalization. The stochastic system will be extended later for m precursors. The system for one precursor is obtained as

Now, consider the probability of more than one event occurring during time At (a very small time interval). During time At, there are four different possibilities for an event for each state defined earlier in the birth and death processes. Let [An, Ac_{1}]^{T} be the change in the populations n and c_{1} in time At, assuming that the changes are approximately normally distributed. Then, the four possibilities for [An, Ac_{1}]^{T} may be defined as

Here, the first event represents a death (capture) of neutrons, the second event is a fission event with -1 + (1 - p)v neutrons produced and p_{1}v delayed neutrons precursors produced, the third event represents a transformation of a delayed neutron precursor to a neutron and the fourth event is a birth of a source neutron. The probabilities of these four events are

where it is assumed that the extraneous source produces neutrons randomly following a Poisson process with intensity q.

Considering the mean and covariance of the change for a small time interval At, we have

Here, we note that
and

assuming a = 1/v, we have b = a/l.

Finally, we get

where

and

Here, the assumption is that the changes are approximately normally distributed and, hence, the result implies

where % , ~ N(0, 1) and B^{1/2} is the square root of the matrix B, where B = B^{1/2} ? B^{1/2}.

As At ^ 0, we will have the following Ito stochastic system:
where

and

Here, W_{1}(t) and W_{2}(t) are Wiener processes.

One may generalize this concept into m precursors. Then, the matrices will be

where

Using a similar approach for m precursors, one may get the following Ito stochastic system:

Equation 10.28 is called the stochastic point kinetic equations for m precursors. In Equation 10.28, we may note that if B = 0, then Equation 10.28 transforms into the standard deterministic point kinetic equations, hence Equation 10.28 can be considered a generalization of the standard point kinetic model.