A random walk in reactor physics and neutron transport
IMRE PÁZSIT
Division of Subatomic and Plasma Physics, Nuclear Engineering Group
Chalmers University of Technology SE-412 96 Göteborg, Sweden
E-mail:imre@chalmers.se

The title of this paper alludes to two different meanings of “random”. First, the phrase “Randomwalk” refers to the fact that I selected, at random, a few topics which I myself found fascinating,surprising, and hence hopefully entertaining, in the hope that the reader will also find thementertaining. The phenomena that will be described and discussed here will reveal some
unexpected features, which in some cases are puzzling or even counter-intuitive, and their
explanation sometimes discloses commonly accepted misbeliefs or misunderstandings. I always
found such cases very intriguing. Inevitably, such subjects do not constitute a continuous story,
rather they are picked randomly, hence the first meaning of the phrase “random walk” in the title.
Curiosities similar to the types that will be discussed in this note are usually published as a
“Letter to the Editor” or a “Technical Note”, since they do not contain new research results. A few
examples are given in Ref [1] (meaning of the flux) and Refs [2] - [4] (number of collisions until
slowing down). The readers are encouraged to check up these letters or technical notes. Many are,in contrast to the present article, quite short, often only one page, hence the “output/input ratio” inintellectual entertainment is quite high. I can also recommend the readers to watch out for such
short notes by themselves (although, sadly, the number of such notes seems to be decreasing).
The second reason why the word “random” appears in the title is because the curious facts andphenomena which will be discussed here concern the randomness of neutron transport, manifesting
itself in the fact that the number of neutrons in the system, or the number of detector counts during
a time period, is a random number or random process (hence often referred to as neutron
fluctuations or neutron noise). Random processes in general, whether about neutrons or otherprocesses, have themselves fascinating and surprising properties. The subjects discussed in thissmall essay will hopefully also expedite a wider understanding of the properties and use of neutron
fluctuations in nuclear systems.
With this introduction, I invite the reader to follow me on the random walk in the fascinatingworld of random particle transport.
The ubiquity and importance of fluctuations
Before turning to reactor physics and neutron fluctuations, a few words about randomness andfluctuations in general. The randomness treated in this essay is not related to the intrinsicallystochastic description of quantum mechanics of individual particle processes, where everything
can only be formulated in terms of the so-called wave function, which evolves deterministically,
but which only has a probabilistic interpretation. We will deal with macroscopic classical systems,
which usually have a very large number of freedom (such as a system of 1024 particles), where it
is impossible to specify each variable with the desired accuracy, hence a statistical description isnecessary. Thus, we cannot tell the outcome of the energy loss of neutrons in the laboratory system
for an individual collision, only over a large number of collisions. In some cases quantum
mechanics also plays a role, such as whether a reaction will be absorption or fission, and in thelatter how many neutrons will be released. However, this fact will not require a quantum
mechanical description, only the knowledge of the probabilities or probability distributions of the-1-

corresponding processes and variables (such as the probability distribution of the number ofneutrons in fission).
Since most systems in the macroscopic (hence observable) world are many-body systems, withsome exaggerations one can say that everything in the world is random - but often we do not “see”
the randomness (the fluctuations are too small). The fluctuations are invisible because themeasurement does not have the proper resolution. Once we have a sufficiently sensitive equipment,
we can see that most everyday processes (such as the temperature in a well-controlled isolatedroom) shows tiny, but definite fluctuations around the mean value.
The randomness, or the fluctuations, are often considered a nuisance, and this is expressed withthe word “noise” which, in the world of acoustics, is surely a phenomenon that with the random
distribution of its frequencies disturbs the enjoyment of music, consisting only of discrete
frequencies. However, more often than not, through their rich information content, thesefluctuations are very useful, and in biological processes even necessary for the existence of life.
My favourite example to support this last statement is the heart rate variability, which is basedon the ECG (electrocardiogram) signals that can be easily measured. For a healthy person, theylook periodical, such as shown in Fig. 1.

Fig. 1 Heartbeat signals

The heartbeats are characterized by the large peaks (denoted in medical praxis with the letter“R”), and the time between two beats, the “interbeat intervals” are denoted in the figure as RR1,
RR2 etc. Of course, a large variation of the interbeat intervals is an indication of heart disorder, but
the opposite is not true “ad infinitum”. Namely, one would be tempted to think that for a person at
total rest, these intervals are perfectly equal, and the healthier a person is, the more constant theinterbeat intervals are. The “perfectly healthy” person in this concept would be one whose RR
interval periods are exactly the same. However, a doctor would say that such a person is already“dead”, because the interbeat intervals must fluctuate with all living organisms. In the terminology
of a lay mind, one could say that the interbeat intervals must vary such that the heart is “training”
to be able to adjust the heart’s beat to meet a change in the physical load (running, lifting weightsetc.).

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Whatever the explanation of these fluctuations, plotting the RR intervals as a function of thebeat number, one apparently gets a totally random process, seemingly without any structure (seethe right hand side of Fig. 2).

Fig. 2. Heartbeat signals (left) and a sequence of interbeat time intervals (right), showing a random behaviour

However, a wavelet analysis readily reveals aself-similar structure (see Fig. 3 to the right).
The curve in the top is the variation of theinterbeat signals, the lower two figures show
their wavelet transforms, which indicates a selfsimilar structure: the lowermost figure is anenlarged view of the area in the red box in the
middle figure, which has the same structure as
the whole of the middle figure. Such a selfsimilar structure exhibits a fractal property,
with a corresponding fractal dimension.
Experience shows that for healthy persons,
there is one single fractal dimension associatedwith the heartbeat data; however, the fractal
structure of the interbeat intervals for persons
with incipient heart failures changes, e.g.
develops a bi-fractal structure, which can beused both for early detection of beginning heart
failures, as well as to determine the type of heart
failure. The analysis of heartbeat signals andtheir use of diagnostics is not very much unlikeof that of analysing process data from nuclear
Fig.3 Interbeat intervals and their wavelet transform
reactors; so much so, that nuclear engineers
have also been involved in analysis of medical
data. At several nuclear engineering conferences there are sessions on analysis of medical signals.
The present author has also endeavoured such a study [5].
One can easily find some intuitive illustrations of the use of the rich information content ofrandom processes. Take as an example birds flying in flocks. The paths of the individual birds
look disordered and random, as they constantly change directions, partly to avoid collisions witheach other. But they succeed in this exceptionally well – birds in a flock do not collide with eachother unintentionally. As a contrast, it often happens that birds are killed by collisions with cars.
On first sight this might feel surprising; a car usually moves on a straight line with constant speed,-3-so one would think it is very easy to predict its trajectory and hence avoid it. But the bird brain
does not work this way; the movement on a straight line with constant velocity has too little
information. A “noisy” movement is much richer in information, and hence the brain of the birds
processes it easier. Of course, in order to do that, they have to be able to interpret the randomcharacter of the movement; but apparently it better suits the brain to process a more complex
information to make predictions than that of a simple process which is poor in information.
Another example is the vision of a bird of prey hovering above the field. It will often discover aprey only when the prey is moving, even if very chaotically. A prey standing still often avoids
being detected, because it sends too little information compared to that of a moving prey.
In what follows, we will see how the more complex character of neutron chains in a multiplying
medium (compared to the emission of neutrons from a radioactive source) help us to extract
information about the system, once we know how to interpret the increased complexity of theunderlying physical process. Soon we will also encounter a puzzling observation, which shows thebeautiful depths of the phenomenon, and that of random processes in general.
Family trees and neutron chains as branching processes
Neutron chains and family trees have much in common what regards their random temporal
evolution. They belong to the category that mathematicians call “branching processes”. The
branching simply means that at random time points some events will occur, in which one entity
will be converted into a random number or new entities (“descendants”). Here the probability of
having at least two descendants must be larger than zero, otherwise the process in not branching.
In this respect “branching” is synonymous with “multiplication”. On the other hand, theprobability for a given event that there will be no descendants can also be larger than zero, whichmeans that the process can also die out. For neutrons, the branching means fission, where one
incoming neutron can induce the emission of a random number of new neutrons, zero also allowed(especially if absorption is also taken into account). In a family tree, the events are not associatedwith time, rather with generations, as will be later seen more in detail. The randomness of thisprocess is expressed in the intuitively clear fact that even if the average of descendants is one, theprocess can still either grow or die out.
This shows that the statistical properties of a branching process are more involved than the caseof neutron emission from a radioactive source. This latter has Poisson statistics, which is“boringly” simple; it is defined by one single parameter, the emission intensity, and hence all
statistical moments (mean, variance etc.) carry the same information (e.g. the variance is equal tothe mean). If we know the mean value, we know everything about the statistics of the process.
This is a consequence of the fact that the individual events (particle emissions) are independent.
The reason for the more complicated, and hence more interesting statistics of a branchingprocess is that its random variables, such as particle numbers, are not independent, rather they arecorrelated (their covariance, i.e. the joint probability of two variables minus the product of theprobability of the two variables, is not zero). Here we already arrived to one of the rather subtle
properties of branching processes which the reader might like to contemplate on. For a neutron
chain, i.e. a collection of all generations started by a single neutron, in each individual fission, thenumber of new neutrons is independent of any other fissions in the system; and for the newneutrons born in fission, their fate is independent from each other in the sense that both the time
for the next fission, as well as the number of new neutrons born in their respective fissions, areindependent from each other. The independence of the development of the individual chains is
actually a basic assumption which we set up for the equations that describe the evolution of theprobability of the number of the neutrons in the system (so called master equations, or ChapmanKolmogorov equations). Yet, due to the fact that they were born simultaneously, will incur-4-temporal correlations in the number of the neutrons at different times in the system. This isobtained from the solution of the master equations, set up for the evolution of the probability.
The apparent contradiction that the independence of the events in the process still leads tocorrelations in the numbers of the entities in the process might be illustrated qualitatively by statingthat although two twins, born simultaneously, will die independently of each other, none of themwill survive the other with a thousand years (or hundreds of generations). Their fates, althoughcontrolled by independent events, will strongly overlap. It is the common birth, the branching,which will create correlations between the number of descendants at a given time, despite that theindividual life events are not affecting each other.
The significance of the correlations in the statistics of random processes, in comparison withthat of the independent processes, obeying a pure Poisson process, can be described as follows.
Positively correlated processes, i.e. when the presence of one entity indicates the increased
likelihood of the presence of another entity (such as in a family tree or a neutron chain), have“over-Poisson” statistics, which simply means that the variance is larger than the mean (the relative
variance is larger than unity). Random processes that are subject to a conservation law, have theproperty that the presence of one entity excludes or diminishes the probability of the presence ofanother entity, show negative correlations. Such processes have a sub-Poisson variance, i.e. thevariance is smaller than the mean.
With this introduction we will now explore some properties of the branching processes by thestory of the mathematical treatment of the extinction of family lines.
The extinction of family names
The question of the extinction of the family names dates back to the early 19 th century and isassociated with the Reverend Henry William Watson, clergyman, mathematician and alpinist, andthe mathematician Francis Galton. The story is very entertainingly described by D. G. Kendall inan essay presented at the 100th anniversary of the London Mathematical Society [6], as well as in
the book of T. E. Harris [7]. Galton became interested in the observation by a certain M. Alphonse
De Candolle (1806-1893) regarding the decay of famous families, “men of note”: peers, judges,
and the like. De Candolle put forth the hypothesis, or question, whether the extinction of the familynames of such noble families is caused by the fact that “a rise in physical comfort and intellectual
capacity is necessarily accompanied by diminution of fertility”.
Galton did not like this conclusion (as I guess most of us having a profession requiring a certain
intellectual capacity would not like either) and asked the help of Watson who formulated the
problem in mathematical terms and gave a solution. Before getting to this, I have to attach theremark that some aspects of the formulation and interpretation of the problem reflect the standardsof society as it was then, with a very different view on gender equality as it is today. For instance,
the extinction of the family names is purely associated with the male line; an assumption that is
not valid any longer.
A simplified version of Watson’s statement of the problem is then as follows:“A large nation, of whom we will only concern ourselves with the adult males, N in number,and who each bear separate surnames, colonise a district. Their law of population is such that, ineach generation, 𝑝0 per cent of the adult males have no male children who reach adult life; 𝑝1 have
one such male child; 𝑝2 have two; and so on. Find (1) what proportion of the surnames will havebecome extinct after r generations; and (2) how many instances there will be of the same surname
being held by m persons.” (An answer by Kendall, with an interesting side-line, will be given
later).

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A simpler task, which Watson solved, is that of the extinction probability q. That is, startingwith one family, what is the probability, that when r goes to infinity, the family becomes extinct?
Instead of giving Watson’s solution, which is somewhat complicated, we will quote here asimpler one, which was provided by Agner Krarup Erlang in 1929. He wrote down what we today
would call a backward type master equation for the extinction probability q. He argued that thisprobability is equal to the probabilities of the mutually exclusive events that: either there will be
no males in the first generation (with probability 𝑝0 ); or there will be one male descendant withprobability 𝑝1 , which then has to die out (with the same extinction probability q that we are lookingfor); or there will be two male descendants with probability 𝑝2 , which then both have to die out;
since they die out independently, this probability is equal to 𝑞2 ; and so on. This can be written as∞2

𝑞 = 𝑝0 + 𝑝1 𝑞 + 𝑝2 𝑞 + ⋯ = ∑ 𝑝𝑛 𝑞𝑛

(1)

𝑛=0

The right hand side of this equation is actually the definition of the generation function of theprobability distribution 𝑝𝑛 . The generating function 𝑔(𝑧) of a discrete probability distribution 𝑝𝑛
is a function of the continuous variable z, which is defined as∞𝑔(𝑧) = ∑ 𝑝𝑛 𝑧 𝑛

(2)

𝑛=0

The generating function can be interpreted as a transform of the discrete probability distribution
𝑝𝑛 into the continuous function 𝑔(𝑧). All generating functions have the property of 𝑔(1) = 1,
which can be easily confirmed from Eq. (2). Further, from the generating function, the momentsof the probability distribution can be easily obtained. For instance, the first moment (expectation)
is obtained as∞〈𝑛〉 ≡ 𝜈 = ∑ 𝑛 𝑝𝑛 = 𝑔′ (𝑧)|𝑧=1

(3)

𝑛=0

With the concept of the generating function, the extinction equation, Eq. (1), can be written as
𝑞 = 𝑔(𝑞)(4)This is a transcendental equation, since 𝑔(𝑞) is a polynomial in q, whose order is equal to the
maximum number of descendants which have a non-zero probability. However, a qualitative
solution, which will actually give an exact quantitative solution with a graphical interpretation,
shown in Fig. 4. The straight line (orange) at 45° represents the left hand side of Eq. (4), and thecurve (blue) represents the right hand side of (4). The solution is the q vale were the two lines
intersect. Here we note that 𝑔(𝑞) is always a convex function, since all coefficients in its Taylor
series are positive (they are the probabilities 𝑝𝑛 ). Since 𝑔(1) = 1, it follows that 𝑞 = 1 (extinction
with 100% certainty) is always a solution of the extinction equation. Interestingly, Watson believed(erroneously) that this is the only solution, i.e. all families will always die out. We will see that
this is not always the case, and is valid only to subcritical or critical processes.
Assuming that 𝑔(0) = 𝑝0 > 0 (otherwise the process never could die out), as the graphicalillustration on Fig. 4 shows, there are two distinct cases. As long as 𝑔′ (𝑧)|𝑧=1 = 𝜈 ≤ 1, 𝜈 being
the average number of male descendants (left hand side of the figure), the only root of theextinction equation (4) is 𝑞 = 1, that is the process (the family line) will die out with 100%
certainty. If 𝜈 > 1, i.e. the average number of male descendants is larger than unity, then theextinction equation has two roots, (right hand side of the Figure); one is less than unity (𝑞 < 1),-6-

the other is the known solution 𝑞 = 1. In this latter case, the relevant solution is the smaller root,𝑞 < 1, i.e. the family will not die out with 100% certainty. (The readers who are curious of theproof why 𝑞 < 1 is the “relevant root”, can find the answer in a recent note (Ref. [8]). This is theroot which Watson overlooked.

Fig. 4. Solution of the extinction equation

The extinction and multiplication of neutron chains in a critical reactor
At this point we switch from families to neutrons and interpret the above in terms of neutron
chains. The only change is that the male descendants will be replaced by neutrons. The above
results say that in a multiplying medium, for 𝜈 ≤ 1, i.e. subcritical and critical systems, a neutron
chain started by a single neutron will die out with 100% certainty, whereas for supercriticalsystems (𝜈 ≥ 1), the probability that the chain will die out is less than 100%.
In the continuation, we will focus on exactly critical systems, with 𝜈 = 1 . There, one notices
an apparent contradiction. In a critical system the expectation of the number of neutrons isconstant. Hence, on the average, a single starting neutron will “live forever”. On the other hand,the above results show that in a critical system, all neutrons will die out with 100% certainty.
But this is not the end of the story yet. It is easy to show (but we abstain from a derivation) thatin a critical system, the variance diverges; it tends to infinity linearly with the number ofgenerations (for families) or with time (for neutron chains). This latter result, i.e. that the varianceincreases linearly with time is also valid to a similar process, the random walk, and in this context
it is called the “Einstein relation”. The derivation for neutron chains can be found in [9] and [10].
The divergence of the variance at criticality has long puzzled the nuclear engineeringcommunity. This would mean that a reactor, operating at critical, would sooner or later experiencea large power excursion. But such a case, a sudden, uncontrolled burst of the neutron flux in a
critical reactor, due to purely probabilistic reasons, has never been observed. In Ref. [9] there is adetailed discussion of the possible reasons. The most obvious is that it is the reactor control, nottaken into account in the above considerations, which prevents large deviations from the desired
flux level. Another suggestion is that a stationary reactor is never critical, only close to critical,
because there are always extraneous neutron sources, not arising from the chain reaction, such ascosmic radiation, decay of fission products etc. And as it can also be easily shown, in a subcritical,
source-driven reactor (a reactor in which the chain reaction is maintained by an extraneous source),
the variance remains finite, and asymptotically constant.
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We will show now that, contrary to the standard belief, there is no need to refer to reactor
control or extraneous sources to explain why divergence at criticality has not been observed, theexplanation lies in the same mathematical theory as the one which creates the apparent
contradictory facts. Let us summarize these facts:
In a critical reactor•

The expectation of the number of neutrons is constant;•

The extinction probability equals unity;•

The variance diverges.

How is this possible at the same time?
Here we arrived at a very crucial fact in probability theory which is often overlooked in neutron
noise theory, i.e. the concept of ergodicity and its applicability. Namely, the laws and relationships
of probability theory are always defined over statistical ensembles, consisting of a large numberof identical objects and corresponding events. The laws say nothing about the behaviour of theindividual. However, in practice, we have very seldom an ensemble of a large number of identicalobjects. We do not have thousands of identical reactors to make an experiment; we have only one
reactor and would like to extract information on its random behaviour based on this only reactor.
Thus, instead of investigating a thousand of identical reactors, we observe our reactor a thousand
different times, in the hope that we obtain the same information.
This is the concept of ergodicity, which means that ensemble averages can be replaced by time
averages. And the concept sounds plausible. For instance, if we want to investigate if a coin is fair,i.e. whether the probability of heads and tails is equal, we can either toss a thousand identical coins,and count the number of coins with heads and tails, or toss the same coin a thousand times, andcount the number of heads and tails. There does not seem to be any difference in the two methods,
except that the latter is much easier, because it requires only one coin.
Indeed, in this case there is no difference in the result; and the reason is that the processdescribed above is ergodic. In the experiment of tossing the coins and counting the heads and tails,
there is no difference between the ensemble average and the time average (average over thenumber of tossing the same coin).
One would be tempted to think that all physically realistic processes are ergodic, and indeed agreat many physical processes are ergodic. But it might come as a surprise how many are not; andin particular how a small change in the definition of a process leads to violating ergodicity. Takefor instance coin tossing again. But instead of counting the number of heads and tails, we now
count the difference between heads and tails, as a function of the number of the tosses. Heads
count as +1, and tails as -1, and if there are more tails than heads, the value of the process isnegative. This is then a discrete random process, equivalent to a one-dimensional random walkwith equal probability of stepping to the left and to the right.
It is easy to prove that this process is not ergodic. A rigorous mathematical proof is not given
here, it can easily be found in the literature. One indication that this process is not ergodic is that
only stationary processes, whose statistical properties are invariant to a time shift, can be ergodic.
For such processes both the mean value and the variance must be constant. But we alreadymentioned that for a random walk, the variance grows linearly with the number of steps. The
constant mean (expectation), together with the diverging variance, is possible through the fact that
if we consider a large number of random walks, half of them will diverge to plus infinity, the otherhalf to minus infinity. This gives a mean value of zero. But this requires an ensemble average, i.e.
we never get this result if we continue to follow one and the same process. To get the correct result,
we must re-start the system.
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One can experience the corollaries of these facts in everyday life. For instance this is the reason
why a number of ball games, where the difference of the number of scores decides who wins, suchas tennis, table tennis, volleyball etc. are played in sets. In tennis, a tournament match goes until
one player wins three sets, usually consisting of 6 games won (except in a tie break), instead of
one single match consisting of 18 games won. This latter would not be fair. One has to break the“random walk” into pieces, i.e. re-starting the process, to get some ensemble average. One actually
made a change about 20 years ago in the rules of table tennis, shifting from 3 sets of 21 points to
4 sets of 11 points, which gives a better ensemble average. The non-ergodicity is part of the reason
why a player, winning one set quite comfortably, can lose the second set with equally as large
margins.
Now getting back to our critical reactor, the explanation of the three “contradicting” properties
above is based on the fact that due to the non-ergodicity of the neutron multiplication in a criticalsystem, it has to be interpreted in terms of ensemble averages, and not time averages, hence they
only describe the behaviour of a large number of reactors. It is not the same reactor in which theneutron population will both die out, will remain constant, and diverge at the same time. Rather,
if we have a large number of critical reactors, in most of them the process will die out, whereas ina small fraction of the reactors the number of the neutrons will reach a very high value. To get theextinction probability unity, have a constant expectation, and a diverging variance, we will haveto take the limit of infinitely many reactors. Then, the fraction of the reactors in which the neutron
population does not die out must go to zero, and in these reactors the neutron population must
diverge; in the rest, i.e. in almost all reactors, the population will die out. It is easy to confirm thatwith such a procedure, all three “contradictory” properties can be fulfilled.
The above shows that the expectation is a value which will never be realised; the neutron
number dies out in almost every system, and it will diverge in a zero set of all reactors. This alsomeans that the fraction of the reactors in which the flux will diverge is negligible; if we select areactor at random, it is almost sure that it is not the one which will diverge. In a loose comparison,
one could say that the chance that “our” reactor would explode by random fluctuations iscomparable with the probability that a person would fall out from an aeroplane by the quantum
mechanical tunnel effect. The main point being that the probability is not zero, but sufficiently lowthat we do not worry about it. According to the above, we should not worry about divergence at
critical; we should worry about extinction, since in almost all critical reactors the flux should dieout. It is more to these cases that the effects named in Ref. [9], i.e. the control system, the fact thatthe reactor is slightly subcritical in the presence of extraneous neutrons etc. have to be counted on.
We close this subject with two quotes. The first concerns the answer to the problem statementof Watson, described in the foregoing, i.e. the statistical behaviour of an ensemble of N family
trees. The answer, for the supercritical case (𝜈 ≡ 𝑚 > 1, and hence 𝑞 < 1) is given in a vivid
description by Kendall [6] which, although mathematically correct, would find some difficulties
to be published in this form today if we read the sentence in italics:“What this tells us is that if a large number N of males all having different surnames colonisea district, and if (females being available as and when required) they each propagate with a finite
average replacement rate m > 1 (with ∑ 𝑘 2 𝑝𝑘 < ∞ ), then after a long time has elapsed about qNof the surnames will have disappeared, while the remainder (1 − 𝑞) 𝑁 will persist forever”.
The second quote reminds to the difference between the statistical behaviour of an ensemble
and that of an individual member of the ensemble, especially for non-ergodic processes. This quote
is also due to Kendall, but the present author found it in a fascinating book with the title “Modern
Mathematics for the Engineer”, published in 1961 [11]. It will not go unnoticed that what was“modern” in 1961 may not sound very modern today, at least regarding computer algorithms, yet
I would recommend to each engineering student to have a look at the book, because it really-9-conforms to the citation, attributed to the famous Hungarian-born mathematician John von
Neumann, claiming that “The purpose of computation is insight, not numbers”. The section fromthe book, citing Kendall’s example, is reproduced here with the permission of McGraw-Hill
Educational:
6. 13 The Example of D. G. Kendall's Taxicab Stand1
At a perfectly balanced taxicab stand, either customers wait for taxis or taxis wait for customers.
Customers and taxis arrive with equal frequencies. If customers are counted + and taxis -, thequeue length may be any integer 0, ±1, ±2, … From a queue of length k, the next change leads
with equal probabilities to k + 1 or k - 1. Thus the successive changes are represented by asymmetric random walk. The expected queue length is 0, but it is easily seen that at each individual
stand the queue length is bound to grow to +∞ or -∞. The zero expectation says nothing about thefluctuations at an individual stand; it assures us merely that, in a large ensemble, for any stand withthousands of taxis waiting in despair for customers there is somewhere a stand with equally manycustomers waiting vainly for a taxi.
It should be borne in mind that in building taxi stands, elevators, etc., we are interested in thefluctuations in time at one particular counter, not in large ensembles balanced in the manner
described. Statistical equilibrium is good where it is really meaningful – e.g., in an ensemble ofmany telephone trunk lines. But little satisfaction can be derived from a judicial statisticalequilibrium where for each innocently condemned person we find a felon running free.
1. Kendall, D. G., Some Problems in the Theory of Queues, J. Roy. Statist. Soc.,
ser. B, vol. 13, pp. 151-185, 1951.

A remark on transport in stochastic media
Kendall’s taxi stand example lends a wonderful illustration and support to the scepticism andreservation of this author concerning the treatment of neutron transport in stochastic media. A
stochastic medium is one whose composition (i.e. space-dependence of the cross sections) cannot
be described with a deterministic function, rather only through probability distributions or density
functions. One practical example of such a medium is the core of a pebble bed reactor, in which
neither the exact position of the pebbles in the core during operation, nor the position of theuranium particles inside the pebbles are known. Another example is neutron transport in geologicalformations, or especially in corium after a core meltdown, where neither the geometry, nor thecomposition is known.
Mathematically, this is a rather fascinating and challenging area. Not surprisingly, it drew a lotof interest among transport theoreticians. There exists a vast literature of journal publications, andan excellent inventory of the methods and results is found in the book of Pomraning [12]. Many
ANS conferences have sessions for “stochastic transport”, which also includes transport instochastic media.
My problem with this topic is not the academic quality of the work, which is undoubtedly ofhigh standard. Rather, the lack of relevance of the results to any practical situation. Assume for
instance, that for a given probability distribution of the random medium, one can show that the
expectation of the reactivity for such systems is zero, so that the reactor is “critical on the average”.
In the parlance of Kendall’s taxi stand problem, this merely tells us that in a large ensemble ofsuch cores, for a subset of deeply subcritical cores there will be somewhere an equally sizedcollection of cores that are prompt supercritical. It does not say anything about our reactor, only
about the statistical average over a large number of reactors with an identical probability
distribution of their core material. This is hardly useful information for a given selected reactor.

-10-The results could be useful if one could do something similar as in the case of stationary ergodic
time-dependent process, where one can replace the ensemble average with time average. However,the process here is not time-, but space-dependent (transport in stochastic media is only treated instatic problems). Hence, such a step would require replacing ensemble averages with spatial
averages. This would require that the system is ergodic in space. But this would only be possible
in infinite media (ergodicity requires stationarity), which is irrelevant for practical applications.
The advantage of time-dependent processes (i.e. where the statistics lies in the time evolutionof sequence of events), is that time flows from minus infinity to plus infinity. Even in the casewhen the process is not ergodic, we can estimate ensemble averages by re-starting the process (orits observation, i.e. the measurement). This way one avoids the need of having a large number ofidentically distributed systems, and useful statistical information can be obtained by observing one
and the same system repeatedly. This is the case with the non-ergodic tennis and table tennis
games, where the process is restarted after each set. The aggregate of the results of several sets
gives an estimate of the ensemble average, without the need of observing a large number ofidentically skilful players competing simultaneously. Similarly, in the Feynman-alpha or varianceto mean method (see the next section), the average and the variance of the number of counts in agiven time period is estimated by making several measurements with the same time length.
However, there is no such possibility with processes varying randomly in space. Therefore, theresults of the corresponding theories, however beautiful and intellectually pleasing they may be,give just as irrelevant information on a given selected reactor core as the expected queue length ata taxi stand.
The utilisation of neutron fluctuations in reactors
How, then, can one utilise the stochastic nature of the neutron population in nuclear engineering?
From the reasoning so far, we might conclude that 1) the neutron fluctuations in a reactor (or any
system containing fissile material) carry useful, non-trivial information on the system, and 2) weneed to be careful how we apply our methods, when we only have access to measurement datafrom one reactor, and not from an ensemble of identical reactors.
One example where the random character of neutron transport and multiplication is utilised is
the Monte Carlo method. Traditionally, this method was invented as a numerical tool to calculate
expectations (mean values) of the neutron population, i.e. to obtain a solution of the deterministic
neutron transport equation. In the method the fate of a large number of neutrons is followed up, by
simulating the possible events (place of next collision, type of reaction, number of secondary
neutrons etc.), according to the known probability laws, determined by the material andgeometrical properties of the medium, i.e. the reactor core. Ensemble averages are calculated byarranging the simulations into batches, where in each batch the simulations are re-started andrepeated a number of times. The advantage of the Monte-Carlo method is that it is very versatileand effective to treat realistic, inhomogeneous systems with complicated geometries, by keeping
all variables (space, angle, energy), without making approximations.
However, the Monte-Carlo method is not the truly characteristic utilization of the neutron
fluctuations in the spirit described in the previous sections; it is mentioned here only as anillustration of the fact that often, even if only average (“deterministic”) quantities are sought, it
often gives advantages to utilize the fact that the underlying process is random. But the MonteCarlo method does not utilize the information contained in the higher moments of the neutron
distribution. The archetype of the applications which does utilize this information is thedetermination of the subcritical reactivity 𝜌 < 0 of a reactor, in which a stationary neutron
population is maintained by an extraneous neutron source. Since in a subcritical system, there is arelationship between the flux level and the subcritical reactivity, one could think that the reactivity
𝜌 can be determined from the measurement of the number of detector counts Z during a time period-11-

t. However, for this we should need to know exactly the intensity (strength) S of the extraneous
source, the detector efficiency 𝜀, and so on, which are usually not known.
However, since the second moment of a branching process contains independent informationfrom that of the first moment, a more efficient way is to use the relative variance, or variance tomean, of the detector counts Z as a function of the measurement time t. In a simplified form,
neglecting delayed neutrons, the variance to mean of the detector counts is given by the formula𝜎𝑍2 (𝑡)= 1 + 𝑌(𝑡) ≡ 1 + 𝜀 𝐴 (1 −𝑍(𝑡)1−𝑒 −𝛼 𝑡
𝛼𝑡

);

−𝜌

𝛼= Λ

(5)

Here, A is a known nuclear constant, and Λ is the (known) neutron generation time. One can seethat the source S exactly vanishes from the expression, since both the variance and the mean arelinearly proportional to it. The detector efficiency 𝜀 is still present; however, due to the non-linear
dependence of the relative variance on the measurement time, the parameter 𝛼, which contains thesought reactivity 𝜌, can be determined by curve fitting without the knowledge of 𝜀. It is also seenthat the relative variance is over-Poisson (larger than unity), and that the sought information iscontained in the deviation from the Poisson variance.
The variance to mean method is commonly called the Feynman-alpha method, because it was
first suggested, based on heuristic derivations, by R. Feynman and colleagues [13]. The fullformula (containing delayed neutrons) was rigorously derived by L. Pál (1925-2019) in his seminal
work on the stochastic theory of neutron fluctuations, the so-called Pál-Bell equation [14].
Concluding remarks
It is the hope of the author that this small essay will induce some interest in the young readersregarding the subtleties and surprises of random processes in general, and neutron fluctuations inparticular. This latter area (also called “zero power reactor noise”) is under intensive development
both in reactor physics and in nuclear safeguards. The purpose of the latter is to detect, identify
and quantify hidden (e.g. smuggled) fissile materials by non-intrusive methods. This is achievedby using auto- and cross moments of neutrons and gamma photons, emitted from the item due tospontaneous fission, up to the third or fourth order. Neutron fluctuations in power reactors, inducedby random technological processes (boiling of the coolant, flow-induced vibrations of control rods
and fuel assemblies) are used for on-line monitoring of operating power reactors, to detect andidentify incipient failures at an early stage. A review of both zero power and power reactor noise
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