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GENERALISATION OF THE PROBLEM
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Contents 2.1 Joint Distribution of Particle Numbers at Different Time Instants 2.2 Branching Process with Two Particle Types 2.3 Extinction and Survival Probability
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2.1 Joint Distribution of Particle Numbers at Different Time Instants Let t1 = t and t2 = t + u be two time instants where u ≥ 0. Let n(t1 ) and n(t2 ) denote the number of particles present in a multiplying system at the time instants t1 and t2 , respectively. Then, P{n(t2 ) = n2 , n(t1 ) = n1 |n(0) = 1} = p2 (n1 , n2 , t1 , t2 )
(2.1)
is the probability that in the multiplying system, n1 particles are present at t1 and n2 particles at t2 ≥ t1 , provided that one particle existed in the system at t = 0. It is customary to call this description the two-point model. We will now prove the following important theorem.
Theorem 8. In the case of a homogeneous process, the generating function g2 (z1 , z2 , t1 , t2 ) =
∞ ∞
p2 (n1 , n2 , t1 , t2 )z1n1 z2n2
(2.2)
n1 =0 n2 =0
satisfies the functional equation g2 (z1 , z2 , t1 , t2 ) = g[z1 g(z2 , t2 − t1 ), t1 ],
(2.3)
in which g(z, t) is the solution of (1.29) or its equivalent (1.30).
Proof. Since one has P{n(t2 ) = n2 , n(t1 ) = n1 |n(0) = 1} = P{n(t2 ) = n2 |n(t1 ) = n1 }P{n(t1 ) = n1 |n(0) = 1}, Neutron fluctuations ISBN-13: 978-0-08-045064-3
© 2008 Elsevier Ltd. All rights reserved.
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and since each of the n1 particles found in the system at time t1 will start a branching process independently from the others, one can write that ∞ ∞
P{n(t2 ) = n2 , n(t1 ) = n1 |n(0) = 1}z2n2 z1n1 =
n1 =0 n2 =0
∞ ∞
P{n(t2 ) = n2 |n(t1 ) = n1 }z2n2
n1 =0 n2 =0
×P{n(t1 ) = n1 |n(0) = 1}z1n1 =
∞
[z1 g(z2 , t2 − t1 )]n1 P{n(t1 ) = n1 |n(0) = 1}.
n1 =0
This proves the theorem, as expressed by equation (2.3). As a generalisation of the case, let us now determine the generating function of the probability distribution P{n(tj ) = nj , j = 1, . . . , k|n(0) = 1} = pk (n1 , . . . , nk , t1 , . . . , tk ). Introduce the notations t1 = t and tj − tj−1 = uj−1 , j = 2, . . . , k. By using these, the relation (2.3) can be written in the form g2 (z1 , z2 , t, u1 ) = g[z1 g(z2 , u1 ), t]. Defining the operation ˆ j zj = zj g(zj+1 , uj ), G (2.4) one notes that the expression g[z1 g(z2 , u1 ), t] can be considered as the transform of the function g1 (z1 , t) = g(z1 , t) that results from the operation ˆ 1 z1 = z1 g(z2 , u1 ) G applied directly to the variable z1 . By using (2.4), one can write that ˆ j+1 G ˆ j zj = zj g(G ˆ j+1 zj+1 , uj ) = zj g[zj+1 g(zj+2 , uj+1 ), uj )], G by the virtue of which one has ˆ 2G ˆ 1 z1 , t) = g{z1 g[z2 g(z3 , u2 ), u1 , t]}. g3 (z1 , z2 , z3 , t, u1 , u2 ) = g(G By induction, one arrives at ˆ j−1 · · · G ˆ 1 z1 , t), gj (z1 , . . . , zj , t, u1 , . . . , uj−1 ) = g(G
(2.5)
where g1 (z1 , t) = g(z1 , t) and j = 2, . . . , k. Expression (2.5) is the generating function of the j-point model.
2.1.1 Autocorrelation function of the particle number In many cases, one needs the autocorrelation function E{[n(t) − m1 (t)][n(t + u) − m1 (t + u)]} = Rn,n (t + u, t) which will now be calculated. Actually, it would be more correct to call the function Rn,n (t + u, t) the autocovariance function. However, whenever it does not lead to confusion, following common practice, Rn,n will be referred to as the autocorrelation.
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First, determine the expectation E{n(t)n(t + u)} =
∂2 g2 (z1 , z2 , t, u) ∂z1 ∂z2
. z1 =z2 =1
Introducing the notation s = s(z1 , z2 ) = z1 g(z2 , u) and considering that ∂g(s, t) dg ∂s = , ∂z1 ds ∂z1 one can write down the relation
as well as
∂g(s, t) dg ∂s = , ∂z2 ds ∂z2
d 2 g ∂s ∂s dg ∂2 s ∂ 2 g2 = 2 + . ∂z1 ∂z2 ds ∂z1 ∂z2 ds ∂z1 ∂z2 From this it follows that E{n(t)n(t + u)} = [m2 (t) + m1 (t)]m1 (u). After a short calculation, one obtains E{[n(t) − m1 (t)][n(t + u) − m1 (t + u)]} = Rn,n (t + u, t) = D2{n(t)}e αu ,
(2.6)
(2.7)
in which, according to (1.60) D2{n(t)} =
[Q qα2 − 1]e αt (e αt − 1), Qq2 t,
if α = 0, if α = 0.
It is seen that D2{n(0)} = 0, and if α = −a < 0,
then
lim D2{n(t)} = 0,
t→∞
hence lim Rn,n (t + u, t) = 0.
t→∞
The normalised covariance function Cn,n (t + u, t) =
Rn,n (t + u, t) D{n(t)} αu = e D{n(t)}D{n(t + u)} D{n(t + u)}
(2.8)
is called the (traditional) autocorrelation function of n(t). In a critical system, i.e. when α = 0, one has t , Cn,n (t + u, t) = t+u and further that n(t) and n(t + u) − n(t) are uncorrelated for every finite t and u. This follows from (2.7), since for α = 0 E{[n(t) − m1 (t)][n(t + u) − m1 (t + u)]} − E{[n(t) − m1 (t)]2 } = 0, i.e. E{[n(t) − m1 (t)][n(t + u) − n(t)]} = 0. This observation implies that in a subcritical system near to criticality, the variation in the number of particles (increase or decrease) under a time period u following a time instant t is not correlated with the number of particles at t. This statement may have important consequences regarding the characteristics of the fluctuations in quasi-critical systems.
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2.2 Branching Process with Two Particle Types Suppose that in a multiplying system, two different particle types, which can be converted into each other, develop a branching process. Again, it is assumed that a particle of any type found in the multiplying system at any given time can induce a reaction and generate a number of progeny independently from its own past and from the history of the other particles present. Denote the two particle types as T1 and T2 , respectively. Let n1 (t) be the number of particles of type T1 and n2 (t) that of the particles of type T2 in the multiplying system at t ≥ 0, respectively. Further, let Qi t + o(t), i = 1, 2 be the probability that during the time period t → 0, a particle of type Ti (i = 1, 2) will induce a reaction. As a result of the reaction, ν particles of type T1 and µ particles of type T2 will be generated.1 Let (i)
P{ν = k, µ = l|Ti } = fk,l
(2.9)
denote the probability that in a reaction induced by one particle of the type Ti , the number of generated particles of types T1 and T2 will be k and l, respectively. The normalisation condition reads as ∞ ∞
(i)
fk,l = 1,
i = 1, 2.
k=0 l=0
Define the probabilities P{n1 (t) = n1 , n2 (t) = n2 |n1 (0) = 1, n2 (0) = 0} = p(1) (n1 , n2 , t)
(2.10)
P{n1 (t) = n1 , n2 (t) = n2 |n1 (0) = 0, n2 (0) = 1} = p(2) (n1 , n2 , t).
(2.11)
and 1 , n2 , t) is the probability that at time t ≥ 0 there will be n1 particles of type T1 and n2 particles of type T2 in the system, provided that at t = 0 there was one particle of type Ti and no particle of the opposite type in the system. Introduce the generating functions
Obviously, p(i) (n
g (z1 , z2 , t) = (i)
∞ ∞
p(i) (n1 , n2 , t)z1n1 z2n2
i = 1, 2
(2.12)
n1 =0 n2 =0
and q(i) (z1 , z2 ) =
∞ ∞
(i)
fk,l z1k z2l ,
i = 1, 2.
(2.13)
k=0 l=0
The g (i) and q(i) fulfil the conditions g (i) (1, 1, t) = 1,
q(i) (1, 1) = 1
and g (i) (z1 , z2 , 0) = zi ,
i = 1, 2.
The backward Kolmogorov equation will now be derived for the probabilities p(1) (n1 , n2 , t) 1 Naturally
ν and µ are random variables.
and
p(2) (n1 , n2 , t).
(2.14)
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To be able to follow closely the evolution of the process, the intuitive method which was already used in Section 1.2.1 will be followed. Then one can write t (1) p(1) (n1 , n2 , t) = e −Q1 t δn1 1 δn2 0 + Q1 e −Q1 (t−t ) [ f0,0 δn1 0 δn2 0 0
+ S (1) (n1 , n2 , t ) + S (2) (n1 , n2 , t ) + S (1,2) (n1 , n2 , t )]dt , where S (1) (n1 , n2 , t ) =
∞ k=1
S (2) (n1 , n2 , t ) =
(1)
fk,0
∞ l=1
k
p(1) (au , bu , t ),
a1 +···+ak =n1 b1 +···+bk =n2 u=1
(1)
f0,l
l
p(1) (av , bv , t ),
a1 +···+al =n1 b1 +···+bl =n2 v=1
and S (1,2) (n1 , n2 , t ) =
∞ ∞
(1)
k=1 l=1
×
(1)
fk,l
(1)
(2)
(2)
(1)
(1)
(2)
(2)
a1 +···+ak +a1 +···+al =n1 b1 +···+bk +b1 +···+bl =n2
l k
p(1) (au(1) , bu(1) , t )p(2) (av(2) , bv(2) , t ).
u=1 v=1
Based on this, for the generating function g (1) (z1 , z2 , t), one obtains the following equation: g (1) (z1 , z2 , t) = e −Q1 t z1 + Q1
t
e −Q1 (t−t )
0
∞ ∞
fk,l [g (1) (z1 , z2 , t )]k [g (2) (z1 , z2 , t )]l dt , (1)
k=0 l=0
which – by considering the definition (2.13) – can be written in the following form: t g (1) (z1 , z2 , t) = e −Q1 t z1 + Q1 e −Q1 (t−t ) q(1) [g (1) (z1 , z2 , t ), g (2) (z1 , z2 , t )]dt .
(2.15)
0
In a completely analogous way, one can derive the generating function equation t (2) −Q2 t g (z1 , z2 , t) = e z2 + Q 2 e −Q2 (t−t ) q(2) [g (1) (z1 , z2 , t ), g (2) (z1 , z2 , t )]dt .
(2.16)
0
Differentiating with respect to t, from these equations one arrives at ∂g (i) = Qi [q(i) ( g (1) , g (2) ) − g (i) ], ∂t
i = 1, 2.
(2.17)
By introducing the notations s(i) (z1 , z2 ) = Qi [q(i) (z1 , z2 ) − zi ],
i = 1, 2,
(2.18)
the basic equations can be written in a rather simple form as ∂g (i) = s(i) ( g (1) , g (2) ), ∂t together with the initial conditions g (i) (z1 , z2 , 0) = zi , i = 1, 2.
i = 1, 2,
(2.19)
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Following the method used for proving Theorem 2 of Section 1.2.1, the generating function equations corresponding to the forward Kolmogorov equation can also be derived as ∂g (i) ∂g (i) = Qj [q(j) (z1 , z2 ) − zj ] , ∂t ∂zj j=1 2
i = 1, 2.
(2.20)
By using the notation (2.18), the above can be written in the following concise form: ∂g (i) ∂g (i) s(j) (z1 , z2 ) = , ∂t ∂zj j=1 2
i = 1, 2,
(2.21)
together with the initial conditions g (i) (z1 , z2 , 0) = zi , i = 1, 2. These equations can easily be derived also by rigorous methods, by utilising basic theorems on the properties of the branching processes. An elegant example of such a derivation was given by Sevast’yanov [24].
2.3 Extinction and Survival Probability If a branching process starting from n(0) = 1 takes the value n(t) = 0 for some time t > 0, then it is said to have become degenerate by time t > 0, or in other words the particle population died out by t > 0. The probability P{n(t) = 0|n(0) = 1} = p0 (t) is called the probability of extinction until time t > 0. The probability p0 (t) has already been calculated for the fundamental process when q(z) is a quadratic function of its argument.2 We shall now investigate the properties of extinction in the case of an arbitrary regular process. In the case of a continuous branching process, let At1 denote the event n(t1 ) = 0. From the event At1 it then follows for all t2 > t1 that n(t2 ) = 0, hence it is obvious that At1 ⊆ At2 , i.e. P(At1 ) ≤ P(At2 ). Accordingly, p0 (t1 ) ≤ p0 (t2 ), if t1 ≤ t2 . It is seen that the probability of the extinction p0 (t) occurring until time t > 0 is a non-decreasing function of t. From this it follows that max p0 (t) = lim p0 (t) = p ≤ 1, t→∞
0
thus it is reasonable to call the limit value lim p0 (t) = p
t→∞
(2.22)
the probability of extinction. In the case of a discrete branching process, also called a Galton–Watson process, let Ak be the event that the number of particles in the system is zero at the k-th discrete time point, i.e. n(kt) = nk = 0 where kt = tk is the k-th discrete time point. Let A denote the event of extinction and p = P(A), the probability of the extinction, respectively. Obviously, the event A will occur if at least one of the events Ak , k = 1, 2, . . . occurs, that is. ∞ A= Ak . k=1 2 See
(2.14) and Fig. 1.5.
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Since Ak ⊆ Ak+1 , from the so-called ‘summation theorem’ [17] one has lim P(Ak ) = P(A),
k→∞
and this is equivalent with the relationship lim p0 (kt) = p.
k→∞
In the following, a branching process will be called extincting or degenerate if its extinction probability is equal to 1, and surviving or non-degenerate if its extinction probability is less than 1, respectively. The following fundamental theorem will be proved now. Theorem 9. If q1 ≤ 1 then, for both discrete and continuous processes, the extinction probability p is equal to the trivial solution p = 1 of the equation p = q(p); whereas if q1 > 1, then it is equal to the only non-negative solution of the equation p = q(p) which is less than unity. Proof. First it will be shown that for both the continuous and the discrete case, the extinction probability is determined by the equation p = q( p), after which Theorem 9 will be proved. For continuous processes, with the substitution g(0, u) = p0 (u), from equation (1.23) one obtains p0 (t + u) = g[p0 (u), t]. In view of the limit relation
lim p0 (u) = lim p0 (t + u) = p,
u→∞
u→∞
one has p = g( p, t) for all t, hence from equation (1.29) it follows that p = q( p). For discrete processes, on the other hand, equation (1.73) can be written in the following form: g(z, t) = h[g(z, t − 1)],
t = 1, 2, . . . ,
where g(z, 0) = z
and
h(z) = z + [q(z) − z]W .
Considering that g(0, t) = p0 (t), one has p0 (t) = h[p0 (t − 1)],
where p0 (0) = 0.
Since lim p0 (t) = lim p0 (t − 1) = p,
t→∞
t→∞
the expression p = h(p) = p + [q(p) − p]W is obtained, which is exactly the equation p = q(p). Since q(z) is a probability generating function, i.e. q(1) = 1, the following theorem is valid:3 Theorem 10. If q1 ≤ 1 then for all points 0 ≤ z < 1 one has q(z) > z and the equality p = q(p) holds only for the point z = p = 1; conversely, if q1 > 1, then there exists one and only one point z0 = p < 1 at which q(p) = p; and further, q(z) > z if 0 ≤ z < p, and q(z) < z if p < z < 1, respectively. Proof. By the foregoing steps, the statement on the extinction probability in Theorem 9 has been proved. 3 The
proof of the theorem is given in Appendix A.
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2.3.1 Asymptotic forms of the survival probability Investigate the behaviour of the survival probability P{n(t) > 0|n(0) = 1} = 1 − p0 (t) = R(t)
(2.23)
in the case when t → ∞.
Subcritical medium The following statement appears to be nearly self-evident. Theorem 11. In the case of a continuous process in a subcritical system, the boundedness of the expression 0
1
q(1 − x) + q1 x − 1 dx = − log R0 x[q(1 − x) + x − 1]
(2.24)
is a necessary and sufficient condition for the asymptotic expression R(t) = R0 e −Q(1−q1 )t [1 + o(1)]
(2.25)
to hold when t → ∞. Proof. Substituting the expression g(0, t) = 1 − R(t) into equation (1.29), one arrives at dR = −Q[q(1 − R) + R − 1], dt noting that R(0) = 1. The above has the implicit solution
1
Qt =
R(t)
dx . q(1 − x) + x − 1
One notes that log
1 e −Q(1−q1 )t dx = Q(q1 − 1)t − log R(t) = (q1 − 1) R(t) R(t) q(1 − x) + x − 1 1 1 dx q(1 − x) + q1 x − 1 = dx = k(t). + R(t) x R(t) x[q(1 − x) + x − 1]
From this one immediately obtains the expression R(t) = e −k(t) e −Q(1−q1 )t . Considering that for sufficiently large t values one has R(t) << 1, one observes that e −k(t) = R0 [1 + o(1)], where
R0 = exp − 0
from which (2.24) is immediately obtained.
1
q(1 − x) + q1 x − 1 dx , x[q(1 − x) + x − 1]
(2.26)
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Theorem 12. In a subcritical system with a discrete process, fulfilment of the inequality E{n(1) log n(1)} < ∞
(2.27)
is the necessary and sufficient condition for the asymptotic relation R(t) = R0 [1 − W (1 − q1 )]t [1 + o(1)],
0 < R0 < ∞
(2.28)
to hold when t → ∞, where the probability W is defined in Section 1.4. The proof of this statement is rather long, and it does not involve physical considerations. Hence the proof is given in Appendix B.
Critical medium Determine how the survival probability behaves in a critical medium for t → ∞. Theorem 13. In a critical medium, for a continuous process the asymptotic form of the survival probability R(t) is given by the formula 2 [1 + o(1)], t → ∞ (2.29) R(t) = Qq2 t provided that q2 is finite. Proof. For the proof, start again from (1.29). The substitution g(0, t) = 1 − R(t) yields dR(t) = −Q{q[1 − R(t)] + R(t) − 1}. dt According to the series expansion theorem described in Section A.5, one has 1 q[1 − R(t)] = 1 − q1 R(t) + A[1 − R(t)]R 2 (t), 2 where A[1 − R(t)] ≤ q2 . Since in a critical medium q1 = 1, from the above equation follows that 1 dR(t) = − QA[1 − R(t)]R 2 (t). dt 2 Considering that A[1 − R(t)] = q [θ(t)], where 1 − R(t) ≤ θ(t) < 1, and further that R(t) → 0, if t → ∞, one has A[1 − R(t)] = q2 + φ(t), where φ(t) → 0, if t → ∞. Thus equation (2.30) can be written as 1 dR(t) = − Q[q2 + φ(t)]R 2 (t), dt 2 from which it follows that R(t) =
Q[q2 t +
t 0
2 φ(u)du + C]
.
(2.30)
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Since R(0) = 1, it follows that C = 2, and accordingly R(t) =
−1 t 2 1 2 2 1+ φ(u)du + = + o(1/t), Qq2 t q2 t 0 q2 t Qq2 t
which is exactly what was to be proven. Theorem 14. In the case of a discrete process, the asymptotic form of the survival probability R(t) in a critical medium is given by the formula 2 R(t) = [1 + o(1)], (2.31) Wq2 t in which t tends to infinity on the set of positive real numbers. Proof. The proof will be again based on the series expansion theorem of Section A.5. From the equation R(t + 1) = 1 − h[1 − R(t)], one obtains 1 R(t + 1) = R(t) − WA[1 − R(t)]R 2 (t). 2 Since R(t) → 0 if t → ∞, one has A[1 − R(t)] = q2 + φ(t), where φ(t) → 0, if t → ∞. Thus, 1 R(t + 1) = R(t) − W [q2 + φ(t)]R 2 (t), 2
(2.32)
hence R(t + 1) 1 = 1 − W [q2 + φ(t)]R(t) → 1, R(t) 2 if t → ∞. Rearrange now (2.32) in the following form: 1 R(t + 1) = R(t) − Wq2 R(t)R(t + 1) + ψ(t), 2 where 1 1 ψ(t) = − W φ(t)R 2 (t) − Wq2 R(t)[R(t) − R(t + 1)]. 2 2 Making use of equation (2.32) to substitute the difference R(t) − R(t + 1) yields 1 1 ψ(t) = − W φ(t)R 2 (t) − W 2 q2 [q2 + φ(t)]R 3 (t). 2 4 Considering (2.33), one observes that ψ(t) = 0, R(t)R(t + 1) hence from (2.34), which can be rearranged in the form as lim
t→∞
1 ψ(t) 1 1 = − Wq2 + , R(t) R(t + 1) 2 R(t)R(t + 1)
(2.33)
(2.34)
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it follows that 1 1 1 = + Wq2 + χ(t), R(t + 1) R(t) 2 1 where χ(t) → 0, if t → ∞. Performing the summation t−1 n=0 R(n+1) leads to t−1 n=0
t−1 t−1 1 1 1 = + Wq2 t + χ(n), R(n + 1) R(n) 2 n=0 n=0
and this is nothing else than t−1 1 1 = Wq2 t + χ(n). R(t) 2 n=0
From this the validity of the statement follows directly. The space-dependent theory of extinction is not discussed in the present work. Interesting results of the extinction probability for neutrons in a supercritical sphere were given by Williams [25]. One should also mention that a simple space-dependent theory of the extinction was published by Schroedinger [26] already in 1945.
2.3.2 Special limit distribution theorems It is obvious from the considerations so far that in the case of subcritical and critical systems, pn (t) → 0 if t → ∞ for all n = 1, 2, . . . and it is also clear that ∞ n=1 pn (t) = R(t) → 0 if t → ∞. A more subtle description of the detailed behaviour of the processes for t → ∞ can be investigated with the help of specially defined probabilities.
Asymptotic distribution of the particle number for a surviving process in a subcritical medium Instead of the probability P{n(t) = n|n(0) = 1} of the standard process, let us introduce the probability P{n(t) = n|n(t) > 0} =
pn (t) P{n(t) = n, n(t) > 0} = , P{n(t) > 0} R(t)
∀n ≥ 1,
(2.35)
which corresponds to a process surviving until time t > 0. The following important statement will be proved. Theorem 15. For a subcritical medium, the limit values lim
t→∞
pn (t) = wn , R(t)
n = 1, 2, . . .
(2.36)
do exist, and the quantities w1 , w2 , . . . satisfy the condition ∞
wn = 1,
n=1
hence they can be considered as the probabilities of a regular distribution. Further, the generating function k(z) =
∞ n=1
wn z n
(2.37)
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is given by the formula
k(z) = 1 − exp α
z
0
dx , s(x)
(2.38)
in which α = Q(q1 − 1)
s(x) = Q[q(x) − x].
and
Proof. The proof consists of a series of simple steps. From (2.35) it can be immediately seen that the generating function ∞ P{n(t) = n|n(t) > 0}zn k(z, t) = n=1
can be written in the following form: ∞
k(z, t) =
1 g(z, t) − g(t, 0) 1 − g(z, t) pn (t)zn = =1− R(t) n=1 R(t) R(t)
which, by introducing the notation 1 − g(z, t) = R(z, t),
(2.39)
R(z, t) . R(t)
(2.40)
can be cast into the form k(z, t) = 1 − From (1.29) one has dt =
dg dg = , Q[q(g) − g] s( g)
and from this, one obtains
g(z,t)
t=
du s(u)
g(z,0)
which, by the change of the integration variable u = 1 − x, takes the form t=
1−z
R(z,t)
dx . s(1 − x)
(2.41)
It was earlier pointed out in connection with (2.26) that the equality t=
1 R(t)
also holds, hence
1−z
R(z,t)
dx s(1 − x)
dx = s(1 − x)
1
R(t)
dx . s(1 − x)
Taking into account the equalities
1−z R(z,t)
dx = s(1 − x)
R(t)
R(z,t)
dx + s(1 − x)
1−z R(t)
dx = s(1 − x)
1
R(t)
dx , s(1 − x)
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finally the formula
R(t) R(z,t)
dx = s(1 − x)
1
1−z
dx = s(1 − x)
z
0
du . s(u)
(2.42)
is obtained. Since R(z, t) ≤ R(t) → 0 if t → ∞, the relation s(1 − z) = −αz[1 + (z)] is obviously true in the interval 0 ≤ z < 1 where (z) is continuous, and if z → 0 then (z) → 0. Based on this, the left-hand side of (2.42) can be rearranged as follows: R(t) dx 1 R(z,t) dx 1 R(z, t) = = log , (2.43) α R(t) x[1 + (x)] α[1 + (θ)] R(t) R(z,t) s(1 − x) where R(z, t) ≤ θ ≤ R(t). From (2.42) and (2.43) one obtains z du R(z, t) = exp α[1 + (θ)] , R(t) 0 s(u) and if t → ∞, then
z R(z, t) du lim = exp α . t→∞ R(t) 0 s(u)
Since α < 0,
1
du = ∞, s(u) 0 hence k(1) = 1, from which all statements of the theorem will follow. The following question arises naturally: What is the condition that the expectation of the number of particles in a surviving, subcritical system shall be finite? The answer is given by the theorem below. Theorem 16. The expectation k (1) =
∞
is finite if and only if the integral4 1 αu + s(1 − u) R0 = exp du us(1 − u) 0 n=1 nwn
(2.44)
is finite. In this case the expectation is supplied by the formula lim z↑1
1 dk(z) = , dz R0
(2.45)
and the relationship 1 (2.46) R0 also exists. From this it also follows that the order of the summation and taking the limiting value is interchangeable: lim E{n(t)|n(t) > 0} =
t→∞
∞ n=1 4 This
∞ pn (t) 1 pn (t) = lim = n . t→∞ R(t) t→∞ R(t) R0 n=1
n lim
integral corresponds exactly to the integral (2.24) if it is solved for R0 and one introduces the notation s(z) = Q[q(z) − z].
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Proof. First the statement (2.46) will be proved. It is obvious that E{n(t)} e αt = , P{n(t) > 0} R(t)
E{n(t)|n(t) > 0} =
and since for sufficiently large t one has R(t) ≈ R0 e αt , one immediately obtains (2.46). For the proof of the first part of the statement, based on (2.38) one writes z 1 du dk(z) = −α exp α . dz s(z) 0 s(u) Considering that for values of z slightly less than unity, s(z) ≈ −α(1 − z), one has z 1 1 du −α ≈ = exp , s(z) 1−z 0 1−u and accounting for
z z du du dk(z) ≈ exp α + dz 0 s(u) 0 1−u
yields, after some simple arrangements, the expression 1 αu + s(1 − u) dk(z) ≈ exp du . dz us(1 − u) 1−z From this it follows that dk(z) lim = exp z↑1 dz
0
1
αu + s(1 − u) du . us(1 − u)
Hence, if the integral on the right-hand side is finite, then the expectation derived from the generating function k(z) is also finite, which is just the theorem that was to be proved.
Non-parametric asymptotic distribution in a critical medium An important characteristics of the branching process in a critical medium is that its asymptotic behaviour is exclusively determined by the first two factorial moments of the generating function q(z). It appears to be interesting to construct a random process whose distribution function describes the asymptotic behaviour of the process in a critical medium without material parameters. In the following, we show that the random process n(t) q(t) = (2.47) E{n(t)|n(t) > 0} satisfies this requirement. Theorem 17. If q1 = 1 and q2 is finite, then the distribution function n(t) ≤ x|n(t) > 0 = S(x, t) P E{n(t)|n(t) > 0}
(2.48)
has the property that it converges to the exponential distribution function S(x) = 1 − e −x , if t → ∞.
x ≥ 0,
(2.49)
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Proof. First of all, one notices that E{n(t)|n(t) > 0} ≈
1 Qq2 t. 2
(2.50)
This follows from E{n(t)} R(t) and the fact that in a critical system E{n(t)} = 1, and further that for sufficiently large t one has R(t) ≈ 2/Qq2 t. In order to prove (2.49), define the characteristic function ∞ e −ωx dS(x, t) = E{e −ωq(t) |n(t) > 0} = E{e −ωR(t)n(t) |n(t) > 0} (2.51) ϕ(ω, t) = E{n(t)|n(t) > 0} =
0
which can be written in the following form: ϕ(ω, t) =
R[t, e −ωR(t) ] g[t, e −ωR(t) ] − g(t, 0) =1− . R(t) R(t)
(2.52)
To obtain the asymptotic form of the characteristic function ϕ(ω, t) for t → ∞, an expression for the function R(z, t) for large t values is needed. From (1.29) ∂R(z, t) = −s[1 − R(z, t)] ∂t with the initial condition R(0, z) = 1 − z. By applying the series expansion theorem given in Section A.5, perform now the substitution 1 s(z) = QA(z)(z − 1)2 . 2 Here |A(z)| ≤ q2 and A(z) → q2 if z → 1. After some brief calculation, one arrives at ∂R(z, t) 1 = − QA[1 − R(z, t)]R 2 (z, t). ∂t 2 Taking into account that R(z, t) = 1 − p0 (t) −
∞
pn (t)zn = R(t) −
n=1
the inequality
∞
pn (t)zn ,
n=1
∞ n pn (t)z ≤ 2R(t) |R(z, t)| ≤ R(t) + n=1
holds, from which one has |R(z, t)| ≤ 2R(t) → 0,
if t → ∞.
In view of this, one can write A[1 − R(z, t)] = q2 + δ(z, t), where for all |z| ≤ 1, δ(z, t) tends uniformly to zero if t → ∞. So, equation (2.32) takes the form ∂R(z, t) 1 = − Q[q2 + δ(z, t)]R 2 (z, t), ∂t 2
(2.53)
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whose solution with the initial condition R(z, 0) = 1 − z is given by the equation t 1 1 1 δ(z, u)du . − = Q q2 t + R(z, t) 1 − z 2 0 From this, the formula 2(1 − z) [1 + (z, t)] 2 + Qq2 t(1 − z) immediately follows, in which (z, t) converges uniformly to zero for all |z| ≤ 1 if t → ∞. By performing the substitution R(z, t) =
(2.54)
z = e −ωR(t) in (2.54), from (2.52) one obtains the characteristic function ϕ(ω, t) = 1 − 2
1 − e −ωR(t) (1 + ). R(t)[2 + Qq2 t(1 − e −ωR(t) )]
(2.55)
By utilising the asymptotical formula R(t) =
2 [1 + o(1)], Qq2 t
it follows that 1 ω = , ω+1 ω+1 and this is exactly the characteristic function of the exponential distribution function. lim ϕ(ω, t) = 1 −
t→∞
Asymptotic distribution of the normalised particle number in a supercritical medium In the case of a supercritical medium when q1 > 1 and q2 < ∞, introduce the definition of the normalised particle number n(t) r(t) = = n(t)e −αt , (2.56) E{n(t)} where α = Q(q1 − 1) > 0. Theorem 18. It will be shown that the distribution function r(t) for all u > 0 satisfies the relation E{[r(t + u) − r(t)]2 } = E{r2 (t + u)} − E{r2 (t)} → 0,
if t → ∞.
(2.57)
This means that r(t) converges in quadratic mean, and hence naturally also stochastically, to a random variable denoted by r ∗ , if t → ∞. Proof. For the proof one needs the expectations E{n(t)} = e αt
and
E{n2 (t)} =
Qq2 αt αt e (e − 1) + e αt . α
(2.58)
From these one has E{r(t)} = 1
and
E{r2 (t)} =
Qq2 (1 − e −αt ) + e −αt . α
Naturally, E{r 2 (t)} →
Qq2 , α
if t → ∞.
(2.59)
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Then, one derives the equality E{r(t)r(t + u)} = E{r2 (t)}.
(2.60)
To this order, consider the expectation
∂2 g2 (z1 , z2 , t, u) ∂z1 ∂z2
z1 =z2 =1
= E{n(t + u)n(t)}.
By utilising the expression E{n(t + u)n(t)} = [m2 (t) + m1 (t)]m1 (u) derived in Section 2.1.1, and using the expectations (2.58), one obtains E{n(t + u)n(t)} = E{n2 (t)}e αu . Multiplying by e −α(2t+u) yields E{r(t + u)r(t)} = E{r2 (t)}, from where it follows that E{[r(t + u) − r(t)]2 } = E{r2 (t + u) − r2 (t)}. From this, for every u > 0 the relation E{[r(t + u) − r(t)]2 } = E{r2 (t + u)} − E{r2 (t)} → 0,
if t → ∞
holds. This shows that r(t) converges to the random variable r∗ in quadratic mean (accordingly, also stochastically) if t → ∞. It is known from the theory of stochastic processes that in this case the characteristic function of r(t) converges to the characteristic function of the random variable r∗ if t → ∞, i.e. ∗
ϕ(ω, t) = E{e −ωr(t) } = g(t, exp{−ωe −αt }) → ϕ(ω) = E{e −ωr }. Theorem 19. The characteristic function (Laplace–Stieltjes transform) ∞ e −ωx dS(x) ϕ(ω) = 0
of the limit distribution lim P{r(t) ≤ x} = lim S(x, t) = P{r∗ ≤ x} = S(x)
t→∞
t→∞
is determined by the differential equation s[ϕ(ω)] dϕ(ω) = , dω αω
ϕ(0) = 1,
(2.61)
whose solution in implicit form is given by the equation 1 − ϕ(ω) = ω exp 1
ϕ(ω)
s(x) − α(x − 1) dx . s(x)(x − 1)
Proof. Substitute z with the expression z = exp{−ωe −α(t+u) }
(2.62)
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in the standard relationship g(z, t + u) = g[g(z, t), u]. One obtains ϕ(ω, t + u) = g[ϕ(ωe −αu , t), u], and from this, for t → ∞ the relation ϕ(ω) = g[ϕ(ωe −αu ), u].
(2.63) follows. By utilising the formula derived for the generating function g(z, t) in (1.31), one obtains from (2.63) that ϕ(ω) = ϕ(ωe −αu ) + us[ϕ(ωe −αu )] + o(u), which can also be written in the form o(u) ϕ(ω) − ϕ(ωe −αu ) = s[ϕ(ωe −αu )] + . u u By performing the transition u ↓ 0, the equation dϕ(ω) s[ϕ(ω)] = dω αω is obtained, which corresponds exactly to (2.61). The formula (2.62) will be derived by using a special form of (1.29). Namely, (1.29) can be written in the form dg s(g) − α(g − 1) − dg = αdt g−1 s(g)(g − 1) from which, by integration with respect to t between 0 and t, one obtains g(z,t) s(x) − α(x − 1) dx . 1 − g(z, t) = (1 − z)e αt exp s(x)(x − 1) z With the substitution z = exp{−ωe −αt }, one arrives at 1 − ϕ(ω, t) = (1 − exp{−ωe −αt })e αt exp
ϕ(ω,t)
exp{−ωe −αt }
s(x) − α(z − 1) dx , s(x)(x − 1)
from which (2.62) is immediately obtained if t → ∞, since 1 − exp{−ωe −αt } = ω. t→∞ e −αt lim
This completes the proof of the statements on the asymptotic behaviour of supercritical processes. As an illustration, it seems to be practical to determine the distribution function of S(x) for a quadratic generating function q(z). It will be shown that S(x) = [1 − (1 − p)e −(1−p)x ](x),
(2.64)
where p = 1 − 2(q1 − 1)/q2 is the probability of extinction and (x) is the unit step function being continuous from the right. It can be seen from the definition that S(x) has a first order discontinuity at the point x = 0, since S(−0) = 0, whereas S(+0) = p.
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By using (2.61), the proof of the statement (2.64) goes as follows. Since s(ϕ) = Q f2 (ϕ − 1)(ϕ − p), one has dϕ 1 dω = . (ϕ − 1)(ϕ − p) 1−p ω From here, after integration 1 ϕ(ω) − 1 = C, (2.65) ϕ(ω) − p ω in which the constant C can be determined from the condition ϕ(0) = 1. By applying L’Hospital’s rule, it follows that ϕ(ω) − 1 lim = ϕ (0) = −E{r∗ } = −1, ω→0 ω and based of this, one has C = −1/(1 − p). Then by simple rearrangement one obtains from (2.65) ϕ(ω) =
1 − p + pω . 1−p+ω
(2.66)
From this, the following expression can be deduced for the Laplace transform of the distribution function S(x): ψ(ω) =
ϕ(ω) 1 1−p = − . ω ω ω+1−p
Its inverse corresponds to the formula in (2.64). We note that due to the discontinuity at x = 0 one has dS(x) = pδ(x)dx + (1 − p)2 e −(1−p)x dx,
∀ x ≥ 0.
(2.67)
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Contents 2.1 Joint Distribution of Particle Numbers at Different Time Instants 2.2 Branching Process with Two Particle Types 2.3 Extinction and Survival Probability
36 39 41
2.1 Joint Distribution of Particle Numbers at Different Time Instants Let t1 = t and t2 = t + u be two time instants where u ≥ 0. Let n(t1 ) and n(t2 ) denote the number of particles present in a multiplying system at the time instants t1 and t2 , respectively. Then, P{n(t2 ) = n2 , n(t1 ) = n1 |n(0) = 1} = p2 (n1 , n2 , t1 , t2 )
(2.1)
is the probability that in the multiplying system, n1 particles are present at t1 and n2 particles at t2 ≥ t1 , provided that one particle existed in the system at t = 0. It is customary to call this description the two-point model. We will now prove the following important theorem.
Theorem 8. In the case of a homogeneous process, the generating function g2 (z1 , z2 , t1 , t2 ) =
∞ ∞
p2 (n1 , n2 , t1 , t2 )z1n1 z2n2
(2.2)
n1 =0 n2 =0
satisfies the functional equation g2 (z1 , z2 , t1 , t2 ) = g[z1 g(z2 , t2 − t1 ), t1 ],
(2.3)
in which g(z, t) is the solution of (1.29) or its equivalent (1.30).
Proof. Since one has P{n(t2 ) = n2 , n(t1 ) = n1 |n(0) = 1} = P{n(t2 ) = n2 |n(t1 ) = n1 }P{n(t1 ) = n1 |n(0) = 1}, Neutron fluctuations ISBN-13: 978-0-08-045064-3
© 2008 Elsevier Ltd. All rights reserved.
36
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and since each of the n1 particles found in the system at time t1 will start a branching process independently from the others, one can write that ∞ ∞
P{n(t2 ) = n2 , n(t1 ) = n1 |n(0) = 1}z2n2 z1n1 =
n1 =0 n2 =0
∞ ∞
P{n(t2 ) = n2 |n(t1 ) = n1 }z2n2
n1 =0 n2 =0
×P{n(t1 ) = n1 |n(0) = 1}z1n1 =
∞
[z1 g(z2 , t2 − t1 )]n1 P{n(t1 ) = n1 |n(0) = 1}.
n1 =0
This proves the theorem, as expressed by equation (2.3). As a generalisation of the case, let us now determine the generating function of the probability distribution P{n(tj ) = nj , j = 1, . . . , k|n(0) = 1} = pk (n1 , . . . , nk , t1 , . . . , tk ). Introduce the notations t1 = t and tj − tj−1 = uj−1 , j = 2, . . . , k. By using these, the relation (2.3) can be written in the form g2 (z1 , z2 , t, u1 ) = g[z1 g(z2 , u1 ), t]. Defining the operation ˆ j zj = zj g(zj+1 , uj ), G (2.4) one notes that the expression g[z1 g(z2 , u1 ), t] can be considered as the transform of the function g1 (z1 , t) = g(z1 , t) that results from the operation ˆ 1 z1 = z1 g(z2 , u1 ) G applied directly to the variable z1 . By using (2.4), one can write that ˆ j+1 G ˆ j zj = zj g(G ˆ j+1 zj+1 , uj ) = zj g[zj+1 g(zj+2 , uj+1 ), uj )], G by the virtue of which one has ˆ 2G ˆ 1 z1 , t) = g{z1 g[z2 g(z3 , u2 ), u1 , t]}. g3 (z1 , z2 , z3 , t, u1 , u2 ) = g(G By induction, one arrives at ˆ j−1 · · · G ˆ 1 z1 , t), gj (z1 , . . . , zj , t, u1 , . . . , uj−1 ) = g(G
(2.5)
where g1 (z1 , t) = g(z1 , t) and j = 2, . . . , k. Expression (2.5) is the generating function of the j-point model.
2.1.1 Autocorrelation function of the particle number In many cases, one needs the autocorrelation function E{[n(t) − m1 (t)][n(t + u) − m1 (t + u)]} = Rn,n (t + u, t) which will now be calculated. Actually, it would be more correct to call the function Rn,n (t + u, t) the autocovariance function. However, whenever it does not lead to confusion, following common practice, Rn,n will be referred to as the autocorrelation.
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First, determine the expectation E{n(t)n(t + u)} =
∂2 g2 (z1 , z2 , t, u) ∂z1 ∂z2
. z1 =z2 =1
Introducing the notation s = s(z1 , z2 ) = z1 g(z2 , u) and considering that ∂g(s, t) dg ∂s = , ∂z1 ds ∂z1 one can write down the relation
as well as
∂g(s, t) dg ∂s = , ∂z2 ds ∂z2
d 2 g ∂s ∂s dg ∂2 s ∂ 2 g2 = 2 + . ∂z1 ∂z2 ds ∂z1 ∂z2 ds ∂z1 ∂z2 From this it follows that E{n(t)n(t + u)} = [m2 (t) + m1 (t)]m1 (u). After a short calculation, one obtains E{[n(t) − m1 (t)][n(t + u) − m1 (t + u)]} = Rn,n (t + u, t) = D2{n(t)}e αu ,
(2.6)
(2.7)
in which, according to (1.60) D2{n(t)} =
[Q qα2 − 1]e αt (e αt − 1), Qq2 t,
if α = 0, if α = 0.
It is seen that D2{n(0)} = 0, and if α = −a < 0,
then
lim D2{n(t)} = 0,
t→∞
hence lim Rn,n (t + u, t) = 0.
t→∞
The normalised covariance function Cn,n (t + u, t) =
Rn,n (t + u, t) D{n(t)} αu = e D{n(t)}D{n(t + u)} D{n(t + u)}
(2.8)
is called the (traditional) autocorrelation function of n(t). In a critical system, i.e. when α = 0, one has t , Cn,n (t + u, t) = t+u and further that n(t) and n(t + u) − n(t) are uncorrelated for every finite t and u. This follows from (2.7), since for α = 0 E{[n(t) − m1 (t)][n(t + u) − m1 (t + u)]} − E{[n(t) − m1 (t)]2 } = 0, i.e. E{[n(t) − m1 (t)][n(t + u) − n(t)]} = 0. This observation implies that in a subcritical system near to criticality, the variation in the number of particles (increase or decrease) under a time period u following a time instant t is not correlated with the number of particles at t. This statement may have important consequences regarding the characteristics of the fluctuations in quasi-critical systems.
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2.2 Branching Process with Two Particle Types Suppose that in a multiplying system, two different particle types, which can be converted into each other, develop a branching process. Again, it is assumed that a particle of any type found in the multiplying system at any given time can induce a reaction and generate a number of progeny independently from its own past and from the history of the other particles present. Denote the two particle types as T1 and T2 , respectively. Let n1 (t) be the number of particles of type T1 and n2 (t) that of the particles of type T2 in the multiplying system at t ≥ 0, respectively. Further, let Qi t + o(t), i = 1, 2 be the probability that during the time period t → 0, a particle of type Ti (i = 1, 2) will induce a reaction. As a result of the reaction, ν particles of type T1 and µ particles of type T2 will be generated.1 Let (i)
P{ν = k, µ = l|Ti } = fk,l
(2.9)
denote the probability that in a reaction induced by one particle of the type Ti , the number of generated particles of types T1 and T2 will be k and l, respectively. The normalisation condition reads as ∞ ∞
(i)
fk,l = 1,
i = 1, 2.
k=0 l=0
Define the probabilities P{n1 (t) = n1 , n2 (t) = n2 |n1 (0) = 1, n2 (0) = 0} = p(1) (n1 , n2 , t)
(2.10)
P{n1 (t) = n1 , n2 (t) = n2 |n1 (0) = 0, n2 (0) = 1} = p(2) (n1 , n2 , t).
(2.11)
and 1 , n2 , t) is the probability that at time t ≥ 0 there will be n1 particles of type T1 and n2 particles of type T2 in the system, provided that at t = 0 there was one particle of type Ti and no particle of the opposite type in the system. Introduce the generating functions
Obviously, p(i) (n
g (z1 , z2 , t) = (i)
∞ ∞
p(i) (n1 , n2 , t)z1n1 z2n2
i = 1, 2
(2.12)
n1 =0 n2 =0
and q(i) (z1 , z2 ) =
∞ ∞
(i)
fk,l z1k z2l ,
i = 1, 2.
(2.13)
k=0 l=0
The g (i) and q(i) fulfil the conditions g (i) (1, 1, t) = 1,
q(i) (1, 1) = 1
and g (i) (z1 , z2 , 0) = zi ,
i = 1, 2.
The backward Kolmogorov equation will now be derived for the probabilities p(1) (n1 , n2 , t) 1 Naturally
ν and µ are random variables.
and
p(2) (n1 , n2 , t).
(2.14)
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To be able to follow closely the evolution of the process, the intuitive method which was already used in Section 1.2.1 will be followed. Then one can write t (1) p(1) (n1 , n2 , t) = e −Q1 t δn1 1 δn2 0 + Q1 e −Q1 (t−t ) [ f0,0 δn1 0 δn2 0 0
+ S (1) (n1 , n2 , t ) + S (2) (n1 , n2 , t ) + S (1,2) (n1 , n2 , t )]dt , where S (1) (n1 , n2 , t ) =
∞ k=1
S (2) (n1 , n2 , t ) =
(1)
fk,0
∞ l=1
k
p(1) (au , bu , t ),
a1 +···+ak =n1 b1 +···+bk =n2 u=1
(1)
f0,l
l
p(1) (av , bv , t ),
a1 +···+al =n1 b1 +···+bl =n2 v=1
and S (1,2) (n1 , n2 , t ) =
∞ ∞
(1)
k=1 l=1
×
(1)
fk,l
(1)
(2)
(2)
(1)
(1)
(2)
(2)
a1 +···+ak +a1 +···+al =n1 b1 +···+bk +b1 +···+bl =n2
l k
p(1) (au(1) , bu(1) , t )p(2) (av(2) , bv(2) , t ).
u=1 v=1
Based on this, for the generating function g (1) (z1 , z2 , t), one obtains the following equation: g (1) (z1 , z2 , t) = e −Q1 t z1 + Q1
t
e −Q1 (t−t )
0
∞ ∞
fk,l [g (1) (z1 , z2 , t )]k [g (2) (z1 , z2 , t )]l dt , (1)
k=0 l=0
which – by considering the definition (2.13) – can be written in the following form: t g (1) (z1 , z2 , t) = e −Q1 t z1 + Q1 e −Q1 (t−t ) q(1) [g (1) (z1 , z2 , t ), g (2) (z1 , z2 , t )]dt .
(2.15)
0
In a completely analogous way, one can derive the generating function equation t (2) −Q2 t g (z1 , z2 , t) = e z2 + Q 2 e −Q2 (t−t ) q(2) [g (1) (z1 , z2 , t ), g (2) (z1 , z2 , t )]dt .
(2.16)
0
Differentiating with respect to t, from these equations one arrives at ∂g (i) = Qi [q(i) ( g (1) , g (2) ) − g (i) ], ∂t
i = 1, 2.
(2.17)
By introducing the notations s(i) (z1 , z2 ) = Qi [q(i) (z1 , z2 ) − zi ],
i = 1, 2,
(2.18)
the basic equations can be written in a rather simple form as ∂g (i) = s(i) ( g (1) , g (2) ), ∂t together with the initial conditions g (i) (z1 , z2 , 0) = zi , i = 1, 2.
i = 1, 2,
(2.19)
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Following the method used for proving Theorem 2 of Section 1.2.1, the generating function equations corresponding to the forward Kolmogorov equation can also be derived as ∂g (i) ∂g (i) = Qj [q(j) (z1 , z2 ) − zj ] , ∂t ∂zj j=1 2
i = 1, 2.
(2.20)
By using the notation (2.18), the above can be written in the following concise form: ∂g (i) ∂g (i) s(j) (z1 , z2 ) = , ∂t ∂zj j=1 2
i = 1, 2,
(2.21)
together with the initial conditions g (i) (z1 , z2 , 0) = zi , i = 1, 2. These equations can easily be derived also by rigorous methods, by utilising basic theorems on the properties of the branching processes. An elegant example of such a derivation was given by Sevast’yanov [24].
2.3 Extinction and Survival Probability If a branching process starting from n(0) = 1 takes the value n(t) = 0 for some time t > 0, then it is said to have become degenerate by time t > 0, or in other words the particle population died out by t > 0. The probability P{n(t) = 0|n(0) = 1} = p0 (t) is called the probability of extinction until time t > 0. The probability p0 (t) has already been calculated for the fundamental process when q(z) is a quadratic function of its argument.2 We shall now investigate the properties of extinction in the case of an arbitrary regular process. In the case of a continuous branching process, let At1 denote the event n(t1 ) = 0. From the event At1 it then follows for all t2 > t1 that n(t2 ) = 0, hence it is obvious that At1 ⊆ At2 , i.e. P(At1 ) ≤ P(At2 ). Accordingly, p0 (t1 ) ≤ p0 (t2 ), if t1 ≤ t2 . It is seen that the probability of the extinction p0 (t) occurring until time t > 0 is a non-decreasing function of t. From this it follows that max p0 (t) = lim p0 (t) = p ≤ 1, t→∞
0
thus it is reasonable to call the limit value lim p0 (t) = p
t→∞
(2.22)
the probability of extinction. In the case of a discrete branching process, also called a Galton–Watson process, let Ak be the event that the number of particles in the system is zero at the k-th discrete time point, i.e. n(kt) = nk = 0 where kt = tk is the k-th discrete time point. Let A denote the event of extinction and p = P(A), the probability of the extinction, respectively. Obviously, the event A will occur if at least one of the events Ak , k = 1, 2, . . . occurs, that is. ∞ A= Ak . k=1 2 See
(2.14) and Fig. 1.5.
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Since Ak ⊆ Ak+1 , from the so-called ‘summation theorem’ [17] one has lim P(Ak ) = P(A),
k→∞
and this is equivalent with the relationship lim p0 (kt) = p.
k→∞
In the following, a branching process will be called extincting or degenerate if its extinction probability is equal to 1, and surviving or non-degenerate if its extinction probability is less than 1, respectively. The following fundamental theorem will be proved now. Theorem 9. If q1 ≤ 1 then, for both discrete and continuous processes, the extinction probability p is equal to the trivial solution p = 1 of the equation p = q(p); whereas if q1 > 1, then it is equal to the only non-negative solution of the equation p = q(p) which is less than unity. Proof. First it will be shown that for both the continuous and the discrete case, the extinction probability is determined by the equation p = q( p), after which Theorem 9 will be proved. For continuous processes, with the substitution g(0, u) = p0 (u), from equation (1.23) one obtains p0 (t + u) = g[p0 (u), t]. In view of the limit relation
lim p0 (u) = lim p0 (t + u) = p,
u→∞
u→∞
one has p = g( p, t) for all t, hence from equation (1.29) it follows that p = q( p). For discrete processes, on the other hand, equation (1.73) can be written in the following form: g(z, t) = h[g(z, t − 1)],
t = 1, 2, . . . ,
where g(z, 0) = z
and
h(z) = z + [q(z) − z]W .
Considering that g(0, t) = p0 (t), one has p0 (t) = h[p0 (t − 1)],
where p0 (0) = 0.
Since lim p0 (t) = lim p0 (t − 1) = p,
t→∞
t→∞
the expression p = h(p) = p + [q(p) − p]W is obtained, which is exactly the equation p = q(p). Since q(z) is a probability generating function, i.e. q(1) = 1, the following theorem is valid:3 Theorem 10. If q1 ≤ 1 then for all points 0 ≤ z < 1 one has q(z) > z and the equality p = q(p) holds only for the point z = p = 1; conversely, if q1 > 1, then there exists one and only one point z0 = p < 1 at which q(p) = p; and further, q(z) > z if 0 ≤ z < p, and q(z) < z if p < z < 1, respectively. Proof. By the foregoing steps, the statement on the extinction probability in Theorem 9 has been proved. 3 The
proof of the theorem is given in Appendix A.
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2.3.1 Asymptotic forms of the survival probability Investigate the behaviour of the survival probability P{n(t) > 0|n(0) = 1} = 1 − p0 (t) = R(t)
(2.23)
in the case when t → ∞.
Subcritical medium The following statement appears to be nearly self-evident. Theorem 11. In the case of a continuous process in a subcritical system, the boundedness of the expression 0
1
q(1 − x) + q1 x − 1 dx = − log R0 x[q(1 − x) + x − 1]
(2.24)
is a necessary and sufficient condition for the asymptotic expression R(t) = R0 e −Q(1−q1 )t [1 + o(1)]
(2.25)
to hold when t → ∞. Proof. Substituting the expression g(0, t) = 1 − R(t) into equation (1.29), one arrives at dR = −Q[q(1 − R) + R − 1], dt noting that R(0) = 1. The above has the implicit solution
1
Qt =
R(t)
dx . q(1 − x) + x − 1
One notes that log
1 e −Q(1−q1 )t dx = Q(q1 − 1)t − log R(t) = (q1 − 1) R(t) R(t) q(1 − x) + x − 1 1 1 dx q(1 − x) + q1 x − 1 = dx = k(t). + R(t) x R(t) x[q(1 − x) + x − 1]
From this one immediately obtains the expression R(t) = e −k(t) e −Q(1−q1 )t . Considering that for sufficiently large t values one has R(t) << 1, one observes that e −k(t) = R0 [1 + o(1)], where
R0 = exp − 0
from which (2.24) is immediately obtained.
1
q(1 − x) + q1 x − 1 dx , x[q(1 − x) + x − 1]
(2.26)
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Theorem 12. In a subcritical system with a discrete process, fulfilment of the inequality E{n(1) log n(1)} < ∞
(2.27)
is the necessary and sufficient condition for the asymptotic relation R(t) = R0 [1 − W (1 − q1 )]t [1 + o(1)],
0 < R0 < ∞
(2.28)
to hold when t → ∞, where the probability W is defined in Section 1.4. The proof of this statement is rather long, and it does not involve physical considerations. Hence the proof is given in Appendix B.
Critical medium Determine how the survival probability behaves in a critical medium for t → ∞. Theorem 13. In a critical medium, for a continuous process the asymptotic form of the survival probability R(t) is given by the formula 2 [1 + o(1)], t → ∞ (2.29) R(t) = Qq2 t provided that q2 is finite. Proof. For the proof, start again from (1.29). The substitution g(0, t) = 1 − R(t) yields dR(t) = −Q{q[1 − R(t)] + R(t) − 1}. dt According to the series expansion theorem described in Section A.5, one has 1 q[1 − R(t)] = 1 − q1 R(t) + A[1 − R(t)]R 2 (t), 2 where A[1 − R(t)] ≤ q2 . Since in a critical medium q1 = 1, from the above equation follows that 1 dR(t) = − QA[1 − R(t)]R 2 (t). dt 2 Considering that A[1 − R(t)] = q [θ(t)], where 1 − R(t) ≤ θ(t) < 1, and further that R(t) → 0, if t → ∞, one has A[1 − R(t)] = q2 + φ(t), where φ(t) → 0, if t → ∞. Thus equation (2.30) can be written as 1 dR(t) = − Q[q2 + φ(t)]R 2 (t), dt 2 from which it follows that R(t) =
Q[q2 t +
t 0
2 φ(u)du + C]
.
(2.30)
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Since R(0) = 1, it follows that C = 2, and accordingly R(t) =
−1 t 2 1 2 2 1+ φ(u)du + = + o(1/t), Qq2 t q2 t 0 q2 t Qq2 t
which is exactly what was to be proven. Theorem 14. In the case of a discrete process, the asymptotic form of the survival probability R(t) in a critical medium is given by the formula 2 R(t) = [1 + o(1)], (2.31) Wq2 t in which t tends to infinity on the set of positive real numbers. Proof. The proof will be again based on the series expansion theorem of Section A.5. From the equation R(t + 1) = 1 − h[1 − R(t)], one obtains 1 R(t + 1) = R(t) − WA[1 − R(t)]R 2 (t). 2 Since R(t) → 0 if t → ∞, one has A[1 − R(t)] = q2 + φ(t), where φ(t) → 0, if t → ∞. Thus, 1 R(t + 1) = R(t) − W [q2 + φ(t)]R 2 (t), 2
(2.32)
hence R(t + 1) 1 = 1 − W [q2 + φ(t)]R(t) → 1, R(t) 2 if t → ∞. Rearrange now (2.32) in the following form: 1 R(t + 1) = R(t) − Wq2 R(t)R(t + 1) + ψ(t), 2 where 1 1 ψ(t) = − W φ(t)R 2 (t) − Wq2 R(t)[R(t) − R(t + 1)]. 2 2 Making use of equation (2.32) to substitute the difference R(t) − R(t + 1) yields 1 1 ψ(t) = − W φ(t)R 2 (t) − W 2 q2 [q2 + φ(t)]R 3 (t). 2 4 Considering (2.33), one observes that ψ(t) = 0, R(t)R(t + 1) hence from (2.34), which can be rearranged in the form as lim
t→∞
1 ψ(t) 1 1 = − Wq2 + , R(t) R(t + 1) 2 R(t)R(t + 1)
(2.33)
(2.34)
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it follows that 1 1 1 = + Wq2 + χ(t), R(t + 1) R(t) 2 1 where χ(t) → 0, if t → ∞. Performing the summation t−1 n=0 R(n+1) leads to t−1 n=0
t−1 t−1 1 1 1 = + Wq2 t + χ(n), R(n + 1) R(n) 2 n=0 n=0
and this is nothing else than t−1 1 1 = Wq2 t + χ(n). R(t) 2 n=0
From this the validity of the statement follows directly. The space-dependent theory of extinction is not discussed in the present work. Interesting results of the extinction probability for neutrons in a supercritical sphere were given by Williams [25]. One should also mention that a simple space-dependent theory of the extinction was published by Schroedinger [26] already in 1945.
2.3.2 Special limit distribution theorems It is obvious from the considerations so far that in the case of subcritical and critical systems, pn (t) → 0 if t → ∞ for all n = 1, 2, . . . and it is also clear that ∞ n=1 pn (t) = R(t) → 0 if t → ∞. A more subtle description of the detailed behaviour of the processes for t → ∞ can be investigated with the help of specially defined probabilities.
Asymptotic distribution of the particle number for a surviving process in a subcritical medium Instead of the probability P{n(t) = n|n(0) = 1} of the standard process, let us introduce the probability P{n(t) = n|n(t) > 0} =
pn (t) P{n(t) = n, n(t) > 0} = , P{n(t) > 0} R(t)
∀n ≥ 1,
(2.35)
which corresponds to a process surviving until time t > 0. The following important statement will be proved. Theorem 15. For a subcritical medium, the limit values lim
t→∞
pn (t) = wn , R(t)
n = 1, 2, . . .
(2.36)
do exist, and the quantities w1 , w2 , . . . satisfy the condition ∞
wn = 1,
n=1
hence they can be considered as the probabilities of a regular distribution. Further, the generating function k(z) =
∞ n=1
wn z n
(2.37)
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is given by the formula
k(z) = 1 − exp α
z
0
dx , s(x)
(2.38)
in which α = Q(q1 − 1)
s(x) = Q[q(x) − x].
and
Proof. The proof consists of a series of simple steps. From (2.35) it can be immediately seen that the generating function ∞ P{n(t) = n|n(t) > 0}zn k(z, t) = n=1
can be written in the following form: ∞
k(z, t) =
1 g(z, t) − g(t, 0) 1 − g(z, t) pn (t)zn = =1− R(t) n=1 R(t) R(t)
which, by introducing the notation 1 − g(z, t) = R(z, t),
(2.39)
R(z, t) . R(t)
(2.40)
can be cast into the form k(z, t) = 1 − From (1.29) one has dt =
dg dg = , Q[q(g) − g] s( g)
and from this, one obtains
g(z,t)
t=
du s(u)
g(z,0)
which, by the change of the integration variable u = 1 − x, takes the form t=
1−z
R(z,t)
dx . s(1 − x)
(2.41)
It was earlier pointed out in connection with (2.26) that the equality t=
1 R(t)
also holds, hence
1−z
R(z,t)
dx s(1 − x)
dx = s(1 − x)
1
R(t)
dx . s(1 − x)
Taking into account the equalities
1−z R(z,t)
dx = s(1 − x)
R(t)
R(z,t)
dx + s(1 − x)
1−z R(t)
dx = s(1 − x)
1
R(t)
dx , s(1 − x)
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finally the formula
R(t) R(z,t)
dx = s(1 − x)
1
1−z
dx = s(1 − x)
z
0
du . s(u)
(2.42)
is obtained. Since R(z, t) ≤ R(t) → 0 if t → ∞, the relation s(1 − z) = −αz[1 + (z)] is obviously true in the interval 0 ≤ z < 1 where (z) is continuous, and if z → 0 then (z) → 0. Based on this, the left-hand side of (2.42) can be rearranged as follows: R(t) dx 1 R(z,t) dx 1 R(z, t) = = log , (2.43) α R(t) x[1 + (x)] α[1 + (θ)] R(t) R(z,t) s(1 − x) where R(z, t) ≤ θ ≤ R(t). From (2.42) and (2.43) one obtains z du R(z, t) = exp α[1 + (θ)] , R(t) 0 s(u) and if t → ∞, then
z R(z, t) du lim = exp α . t→∞ R(t) 0 s(u)
Since α < 0,
1
du = ∞, s(u) 0 hence k(1) = 1, from which all statements of the theorem will follow. The following question arises naturally: What is the condition that the expectation of the number of particles in a surviving, subcritical system shall be finite? The answer is given by the theorem below. Theorem 16. The expectation k (1) =
∞
is finite if and only if the integral4 1 αu + s(1 − u) R0 = exp du us(1 − u) 0 n=1 nwn
(2.44)
is finite. In this case the expectation is supplied by the formula lim z↑1
1 dk(z) = , dz R0
(2.45)
and the relationship 1 (2.46) R0 also exists. From this it also follows that the order of the summation and taking the limiting value is interchangeable: lim E{n(t)|n(t) > 0} =
t→∞
∞ n=1 4 This
∞ pn (t) 1 pn (t) = lim = n . t→∞ R(t) t→∞ R(t) R0 n=1
n lim
integral corresponds exactly to the integral (2.24) if it is solved for R0 and one introduces the notation s(z) = Q[q(z) − z].
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Proof. First the statement (2.46) will be proved. It is obvious that E{n(t)} e αt = , P{n(t) > 0} R(t)
E{n(t)|n(t) > 0} =
and since for sufficiently large t one has R(t) ≈ R0 e αt , one immediately obtains (2.46). For the proof of the first part of the statement, based on (2.38) one writes z 1 du dk(z) = −α exp α . dz s(z) 0 s(u) Considering that for values of z slightly less than unity, s(z) ≈ −α(1 − z), one has z 1 1 du −α ≈ = exp , s(z) 1−z 0 1−u and accounting for
z z du du dk(z) ≈ exp α + dz 0 s(u) 0 1−u
yields, after some simple arrangements, the expression 1 αu + s(1 − u) dk(z) ≈ exp du . dz us(1 − u) 1−z From this it follows that dk(z) lim = exp z↑1 dz
0
1
αu + s(1 − u) du . us(1 − u)
Hence, if the integral on the right-hand side is finite, then the expectation derived from the generating function k(z) is also finite, which is just the theorem that was to be proved.
Non-parametric asymptotic distribution in a critical medium An important characteristics of the branching process in a critical medium is that its asymptotic behaviour is exclusively determined by the first two factorial moments of the generating function q(z). It appears to be interesting to construct a random process whose distribution function describes the asymptotic behaviour of the process in a critical medium without material parameters. In the following, we show that the random process n(t) q(t) = (2.47) E{n(t)|n(t) > 0} satisfies this requirement. Theorem 17. If q1 = 1 and q2 is finite, then the distribution function n(t) ≤ x|n(t) > 0 = S(x, t) P E{n(t)|n(t) > 0}
(2.48)
has the property that it converges to the exponential distribution function S(x) = 1 − e −x , if t → ∞.
x ≥ 0,
(2.49)
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Proof. First of all, one notices that E{n(t)|n(t) > 0} ≈
1 Qq2 t. 2
(2.50)
This follows from E{n(t)} R(t) and the fact that in a critical system E{n(t)} = 1, and further that for sufficiently large t one has R(t) ≈ 2/Qq2 t. In order to prove (2.49), define the characteristic function ∞ e −ωx dS(x, t) = E{e −ωq(t) |n(t) > 0} = E{e −ωR(t)n(t) |n(t) > 0} (2.51) ϕ(ω, t) = E{n(t)|n(t) > 0} =
0
which can be written in the following form: ϕ(ω, t) =
R[t, e −ωR(t) ] g[t, e −ωR(t) ] − g(t, 0) =1− . R(t) R(t)
(2.52)
To obtain the asymptotic form of the characteristic function ϕ(ω, t) for t → ∞, an expression for the function R(z, t) for large t values is needed. From (1.29) ∂R(z, t) = −s[1 − R(z, t)] ∂t with the initial condition R(0, z) = 1 − z. By applying the series expansion theorem given in Section A.5, perform now the substitution 1 s(z) = QA(z)(z − 1)2 . 2 Here |A(z)| ≤ q2 and A(z) → q2 if z → 1. After some brief calculation, one arrives at ∂R(z, t) 1 = − QA[1 − R(z, t)]R 2 (z, t). ∂t 2 Taking into account that R(z, t) = 1 − p0 (t) −
∞
pn (t)zn = R(t) −
n=1
the inequality
∞
pn (t)zn ,
n=1
∞ n pn (t)z ≤ 2R(t) |R(z, t)| ≤ R(t) + n=1
holds, from which one has |R(z, t)| ≤ 2R(t) → 0,
if t → ∞.
In view of this, one can write A[1 − R(z, t)] = q2 + δ(z, t), where for all |z| ≤ 1, δ(z, t) tends uniformly to zero if t → ∞. So, equation (2.32) takes the form ∂R(z, t) 1 = − Q[q2 + δ(z, t)]R 2 (z, t), ∂t 2
(2.53)
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whose solution with the initial condition R(z, 0) = 1 − z is given by the equation t 1 1 1 δ(z, u)du . − = Q q2 t + R(z, t) 1 − z 2 0 From this, the formula 2(1 − z) [1 + (z, t)] 2 + Qq2 t(1 − z) immediately follows, in which (z, t) converges uniformly to zero for all |z| ≤ 1 if t → ∞. By performing the substitution R(z, t) =
(2.54)
z = e −ωR(t) in (2.54), from (2.52) one obtains the characteristic function ϕ(ω, t) = 1 − 2
1 − e −ωR(t) (1 + ). R(t)[2 + Qq2 t(1 − e −ωR(t) )]
(2.55)
By utilising the asymptotical formula R(t) =
2 [1 + o(1)], Qq2 t
it follows that 1 ω = , ω+1 ω+1 and this is exactly the characteristic function of the exponential distribution function. lim ϕ(ω, t) = 1 −
t→∞
Asymptotic distribution of the normalised particle number in a supercritical medium In the case of a supercritical medium when q1 > 1 and q2 < ∞, introduce the definition of the normalised particle number n(t) r(t) = = n(t)e −αt , (2.56) E{n(t)} where α = Q(q1 − 1) > 0. Theorem 18. It will be shown that the distribution function r(t) for all u > 0 satisfies the relation E{[r(t + u) − r(t)]2 } = E{r2 (t + u)} − E{r2 (t)} → 0,
if t → ∞.
(2.57)
This means that r(t) converges in quadratic mean, and hence naturally also stochastically, to a random variable denoted by r ∗ , if t → ∞. Proof. For the proof one needs the expectations E{n(t)} = e αt
and
E{n2 (t)} =
Qq2 αt αt e (e − 1) + e αt . α
(2.58)
From these one has E{r(t)} = 1
and
E{r2 (t)} =
Qq2 (1 − e −αt ) + e −αt . α
Naturally, E{r 2 (t)} →
Qq2 , α
if t → ∞.
(2.59)
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Then, one derives the equality E{r(t)r(t + u)} = E{r2 (t)}.
(2.60)
To this order, consider the expectation
∂2 g2 (z1 , z2 , t, u) ∂z1 ∂z2
z1 =z2 =1
= E{n(t + u)n(t)}.
By utilising the expression E{n(t + u)n(t)} = [m2 (t) + m1 (t)]m1 (u) derived in Section 2.1.1, and using the expectations (2.58), one obtains E{n(t + u)n(t)} = E{n2 (t)}e αu . Multiplying by e −α(2t+u) yields E{r(t + u)r(t)} = E{r2 (t)}, from where it follows that E{[r(t + u) − r(t)]2 } = E{r2 (t + u) − r2 (t)}. From this, for every u > 0 the relation E{[r(t + u) − r(t)]2 } = E{r2 (t + u)} − E{r2 (t)} → 0,
if t → ∞
holds. This shows that r(t) converges to the random variable r∗ in quadratic mean (accordingly, also stochastically) if t → ∞. It is known from the theory of stochastic processes that in this case the characteristic function of r(t) converges to the characteristic function of the random variable r∗ if t → ∞, i.e. ∗
ϕ(ω, t) = E{e −ωr(t) } = g(t, exp{−ωe −αt }) → ϕ(ω) = E{e −ωr }. Theorem 19. The characteristic function (Laplace–Stieltjes transform) ∞ e −ωx dS(x) ϕ(ω) = 0
of the limit distribution lim P{r(t) ≤ x} = lim S(x, t) = P{r∗ ≤ x} = S(x)
t→∞
t→∞
is determined by the differential equation s[ϕ(ω)] dϕ(ω) = , dω αω
ϕ(0) = 1,
(2.61)
whose solution in implicit form is given by the equation 1 − ϕ(ω) = ω exp 1
ϕ(ω)
s(x) − α(x − 1) dx . s(x)(x − 1)
Proof. Substitute z with the expression z = exp{−ωe −α(t+u) }
(2.62)
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in the standard relationship g(z, t + u) = g[g(z, t), u]. One obtains ϕ(ω, t + u) = g[ϕ(ωe −αu , t), u], and from this, for t → ∞ the relation ϕ(ω) = g[ϕ(ωe −αu ), u].
(2.63) follows. By utilising the formula derived for the generating function g(z, t) in (1.31), one obtains from (2.63) that ϕ(ω) = ϕ(ωe −αu ) + us[ϕ(ωe −αu )] + o(u), which can also be written in the form o(u) ϕ(ω) − ϕ(ωe −αu ) = s[ϕ(ωe −αu )] + . u u By performing the transition u ↓ 0, the equation dϕ(ω) s[ϕ(ω)] = dω αω is obtained, which corresponds exactly to (2.61). The formula (2.62) will be derived by using a special form of (1.29). Namely, (1.29) can be written in the form dg s(g) − α(g − 1) − dg = αdt g−1 s(g)(g − 1) from which, by integration with respect to t between 0 and t, one obtains g(z,t) s(x) − α(x − 1) dx . 1 − g(z, t) = (1 − z)e αt exp s(x)(x − 1) z With the substitution z = exp{−ωe −αt }, one arrives at 1 − ϕ(ω, t) = (1 − exp{−ωe −αt })e αt exp
ϕ(ω,t)
exp{−ωe −αt }
s(x) − α(z − 1) dx , s(x)(x − 1)
from which (2.62) is immediately obtained if t → ∞, since 1 − exp{−ωe −αt } = ω. t→∞ e −αt lim
This completes the proof of the statements on the asymptotic behaviour of supercritical processes. As an illustration, it seems to be practical to determine the distribution function of S(x) for a quadratic generating function q(z). It will be shown that S(x) = [1 − (1 − p)e −(1−p)x ](x),
(2.64)
where p = 1 − 2(q1 − 1)/q2 is the probability of extinction and (x) is the unit step function being continuous from the right. It can be seen from the definition that S(x) has a first order discontinuity at the point x = 0, since S(−0) = 0, whereas S(+0) = p.
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By using (2.61), the proof of the statement (2.64) goes as follows. Since s(ϕ) = Q f2 (ϕ − 1)(ϕ − p), one has dϕ 1 dω = . (ϕ − 1)(ϕ − p) 1−p ω From here, after integration 1 ϕ(ω) − 1 = C, (2.65) ϕ(ω) − p ω in which the constant C can be determined from the condition ϕ(0) = 1. By applying L’Hospital’s rule, it follows that ϕ(ω) − 1 lim = ϕ (0) = −E{r∗ } = −1, ω→0 ω and based of this, one has C = −1/(1 − p). Then by simple rearrangement one obtains from (2.65) ϕ(ω) =
1 − p + pω . 1−p+ω
(2.66)
From this, the following expression can be deduced for the Laplace transform of the distribution function S(x): ψ(ω) =
ϕ(ω) 1 1−p = − . ω ω ω+1−p
Its inverse corresponds to the formula in (2.64). We note that due to the discontinuity at x = 0 one has dS(x) = pδ(x)dx + (1 − p)2 e −(1−p)x dx,
∀ x ≥ 0.
(2.67)