PHYSICAL REVIEW E 75, 031122 共2007兲
Spectral theory of metastability and extinction in a branching-annihilation reaction
Michael Assaf and Baruch Meerson
Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
共Received 6 December 2006; revised manuscript received 12 February 2007; published 29 March 2007兲
We apply the spectral method, recently developed by the authors, to calculate the statistics of a reactionlimited multistep birth-death process, or chemical reaction, that includes as elementary steps branching A
→ 2A and annihilation 2A → 0. The spectral method employs the generating function technique in conjunction
with the Sturm-Liouville theory of linear differential operators. We focus on the limit when the branching rate
is much higher than the annihilation rate and obtain accurate analytical results for the complete probability
distribution 共including large deviations兲 of the metastable long-lived state and for the extinction time statistics.
The analytical results are in very good agreement with numerical calculations. Furthermore, we use this
example to settle the issue of the “lacking” boundary condition in the spectral formulation.
DOI: 10.1103/PhysRevE.75.031122
PACS number共s兲: 05.40.⫺a, 02.50.Ey, 87.23.Cc, 82.20.⫺w
I. INTRODUCTION
The statistics of rare events, or large deviations, in chemical reactions and systems of birth-death type have attracted a
great deal of interest in many areas of science including
physics, chemistry, astrochemistry, epidemiology, population
biology, cell biochemistry, etc. 关1–14兴. Large deviations become of vital importance when the discrete 共noncontinuum兲
nature of a population of “particles” 共molecules, bacteria,
cells, animals, or even humans兲 drives it to extinction. A
standard way of putting the discreteness of particles into
theory is the master equation 关1,2兴 which describes the evolution of the probability of having a certain number of particles of each type at time t. The master equation is rarely
soluble analytically, and various approximations are in use
关1,2兴. One widely used approximation is the Fokker-Planck
equation which usually gives accurate results in the regions
around the peaks of the probability distribution, but fails in
its description of large deviations—that is, the distribution
tails 关15–17兴. Not much is known beyond the Fokker-Planck
description. In some particular cases 共especially, for singlestep birth-death processes兲 complete statistics, including
large deviations, were determined by applying various approximations directly to the pertinent master equation
关10,12,14,17–23兴. A different group of approaches employs
the generating function formalism 关1,5,8兴; see below. Here
the master equation is transformed into a linear partial differential equation 共PDE兲 for the generating function, and this
PDE is analyzed and solved by various techniques such as
the method of second quantization 关24–26兴 or the more recent time-dependent WKB approximation 关16,27兴. Recently,
we combined the generating function technique with the
Sturm-Liouville theory of linear differential operators and
developed a spectral theory of rare events 关28,29兴. In this
theory the problem of computing the complete statistics of
共not necessarily single-step兲 birth-death systems reduces to
solving an eigenvalue problem for a linear differential operator, the coefficients of which are determined by the reaction
rates.
In this paper we apply the spectral method to the paradigmatic problem of branching A + X → 2X and annihilation X
+ X → E, where A and E are fixed. This multistep single1539-3755/2007/75共3兲/031122共8兲
species birth-death process describes, for example, chemical
oxidation reactions 关18,30兴. If the branching rate is much
higher than the annihilation rate 共the case we will be mostly
interested in throughout the paper兲, a long-lived metastable,
or quasistationary, state exists where the two processes 共almost兲 balance each other. Still, this long-lived state slowly
decays with time, because a sufficiently large fluctuation ultimately brings the system into the absorbing state of no
particles from which there is a zero probability of exiting. In
this type of problems one is interested in the extinction time
statistics and in the complete probability distribution, including large deviations, of the quasistationary state 共formally
defined as the limiting distribution conditioned on nonextinction兲. Turner and Malek-Mansour 关18兴 calculated the mean
extinction time in this system by solving a recursion equation
for the extinction probability. More recently, Elgart and Kamenev 关16兴 reexamined this problem in the light of their
time-dependent WKB approximation for the generating function. Their insightful method readily yields an estimate of the
mean extinction time, but only up to a 共significant兲 preexponential factor. The quasistationary distribution for this system
has not been previously found, and calculating it will be our
objective. In the language of the spectral theory, the mean
extinction time represents the inverse eigenvalue of the
ground state, while the quasistationary distribution is derivable from the ground-state eigenfunction.
The paradigmatic branching-annihilation problem, considered in this paper, has an additional value, as it helps settle
one unresolved issue of the spectral theory. In previous
works 关28,29兴 we considered reactions that conserve parity
of the particles. Parity conservation provides an additional
boundary condition for the PDE for the generating function
which ensures a closed formulation of the problem already at
the stage of the time-dependent PDE. The branchingannihilation process, considered in the present work, does
not conserve parity. As we will show, the “lacking” boundary
condition emerges here 共and in a host of other problems of
this type兲 only at the stage of the Sturm-Liouville theory.
Here is how we organize the rest of the paper. In Sec. II
we apply the spectral method and reduce the governing master equations to a proper Sturm-Liouville problem. In Sec. III
we employ a matched asymptotic expansion to approximately calculate the ground-state eigenvalue and eigenfunc-
031122-1
©2007 The American Physical Society
PHYSICAL REVIEW E 75, 031122 共2007兲
MICHAEL ASSAF AND BARUCH MEERSON
tion and obtain the long-time asymptotics of the generating
function. This asymptotics is used in Sec. IV to extract the
quasistationary probability distribution and compare it with
our numerical results. In Sec. V we calculate the mean extinction time and extinction probability distribution and compare these results with the previous work and with our numerics. Some final comments are presented in Sec. VI.
II. GENERATING FUNCTION AND SPECTRAL
FORMULATION
We consider the branching and annihilation reactions
␮
␭
A→ 2A
and
2A→ 0,
where ␮ , ␭ ⬎ 0 are the rate constants. The 共mean-field兲 rate
equation for the number of particles n共t兲, dn / dt = ␭n − ␮n2
predicts a nontrivial attracting steady state ns = ␭ / ␮ ⬅ ⍀.
Fluctuations invalidate this mean-field result due to the existence of an absorbing state at n = 0. However, when ⍀ 1,
there exists a long-lived fluctuating metastable 共or quasistationary兲 state, which slowly decays in time, implying a slow
growth of the extinction probability. The statistics of this
quasistationary state and of the extinction times are in the
focus of our attention here.
The master equation for the probability Pn共t兲 to find n
particles at time t can be written as
␮
d2Gst
dGst
共1 − x2兲 2 + ␭x共x − 1兲
= 0,
2
dx
dx
must also be bounded at x = −1. Then Eq. 共5兲 immediately
yields a second boundary condition Gst⬘ 共x兲兩x=−1 = 0, where the
prime stands for the x derivative. Combined with Gst共1兲 = 1,
this condition selects the steady-state solution Gst共x兲 = 1 describing an empty state.
Now let us introduce a new function g共x , t兲 = G共x , t兲
− Gst共x兲 = G共x , t兲 − 1 关which obeys Eq. 共4兲 with a homogenous
boundary condition g共x = 1 , t兲 = 0 and is bounded at x = −1兴
and look for separable solutions, gk共x , t兲 = e−␥kt␸k共x兲. We obtain
共1 − x2兲␸k⬙共x兲 + 2⍀x共x − 1兲␸k⬘共x兲 + 2Ek␸k共x兲 = 0,
2⍀␸k⬘共− 1兲 + Ek␸k共− 1兲 = 0,
共1兲
⬁
共2兲
2e−2⍀x共1 + x兲2⍀
,
1 − x2
共9兲
we arrive at an eigenvalue problem of the Sturm-Liouville
theory 关32兴. Once the complete set of orthogonal eigenfunctions ␸k共x兲 and the respective real eigenvalues Ek, k
= 1 , 2 , . . ., are calculated, one can write the exact solution of
the time-dependent problem for G共x , t兲:
n=0
⬁
where x is an auxiliary variable. Once G共x , t兲 is known, the
probabilities Pn共t兲 can be recovered from the Taylor expansion:
冏
共8兲
with the weight function
We introduce the generating function 关1,2,8兴
1 ⳵nG共x,t兲
Pn共t兲 =
n! ⳵ xn
共7兲
for each k = 1 , 2 , . . . . Notice that the eigenvalue Ek enters the
boundary condition. Rewriting Eq. 共6兲 in a self-adjoint form
w共x兲 =
G共x,t兲 = 兺 xn Pn共t兲,
共6兲
where Ek = ␥k / ␮. One boundary condition is of course
␸k共1兲 = 0. The second boundary condition comes from the
demand that ␸k共x兲 be bounded at x = −1. Then Eq. 共6兲 yields
a homogenous boundary condition
n ⱖ 1,
d
P0共t兲 = ␮ P2共t兲.
dt
共5兲
关␸k⬘共x兲exp共− 2⍀x兲共1 + x兲2⍀兴⬘ + Ekw共x兲␸k共x兲 = 0,
␮
d
Pn共t兲 = 关共n + 2兲共n + 1兲Pn+2共t兲 − n共n − 1兲Pn共t兲兴
dt
2
+ ␭关共n − 1兲Pn−1共t兲 − nPn共t兲兴,
of particles. Now, the steady-state solution of Eq. 共4兲,
Gst共x兲 = G共x , t → ⬁ 兲, which obeys the equation
冏
.
G共x,t兲 = 1 + 兺 ak␸k共x兲e−␮Ekt ,
where the amplitudes ak are given by
共3兲
x=0
By virtue of Eqs. 共2兲 and 共3兲, G共x , t兲 must be analytical, at all
times, at x = 0. Equations 共1兲 and 共2兲 yield a single PDE for
G共x , t兲 关16兴:
⳵G ␮
⳵ 2G
⳵G
= 共1 − x2兲 2 + ␭x共x − 1兲
.
⳵t 2
⳵x
⳵x
共4兲
Conservation of probability yields one 共universal兲 boundary
condition for this parabolic PDE: G共1 , t兲 = 1 关31兴. What is the
second boundary condition? Note that G共x = −1 , t兲 must be
bounded at all times, as it is equal to the difference between
the sum of the probabilities to have an even number of particles and the sum of the probabilities to have an odd number
共10兲
k=1
ak =
冕
1
−1
关G共x,t = 0兲 − 1兴␸k共x兲w共x兲dx
冕
.
1
共11兲
␸2k 共x兲w共x兲dx
−1
As all Ek are positive, Eq. 共10兲 describes decay of initially
populated states k = 1 , 2 , . . ., so the system ultimately approaches the empty state G共x , t → ⬁ 兲 = 1. Being mostly interested in the case of ⍀ 1, we note that while the eigenvalues of the “excited states” E2 , E3 , . . . scale like O共⍀兲 1
关33兴, the “ground-state” eigenvalue E1 is exponentially small
关18兴. Therefore, at sufficiently long times ␮⍀t = ␭t 1, the
contribution from the excited states to G共x , t兲 becomes negligible, and we can write
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PHYSICAL REVIEW E 75, 031122 共2007兲
SPECTRAL THEORY OF METASTABILITY AND…
G共x,t兲 = 1 + a1␸1共x兲e−␮E1t .
共12兲
So we need to calculate the ground-state eigenvalue E1, the
eigenfunction ␸1共x兲, and the amplitude a1. 共Actually, the eigenvalue E1 was calculated earlier 关18兴, but we will rederive
it here.兲 Note that, as E1 is exponentially small, the boundary
condition 共7兲 for the ground state reduces, up to an exponentially small correction, to
␸1⬘共− 1兲 = 0.
共13兲
⬁
⌫关2 + j,− 2⍀兴 − ⌫关2 + j,− 2⍀共1 + x兲兴
E1
.
␸b共x兲 ⯝ 1 −
2兺
2⍀ j=0
j ! 共2⍀ + j + 1兲
共18兲
One can check that the perturbative solution in the bulk is
valid 关that is, ␦␸b共x兲 1兴 as long as 1 − x 1 / ⍀.
In the boundary layer 1 − x 1 we can disregard, in Eq.
共6兲, the 共exponentially small兲 last term and arrive at the same
equation as before: 共1 + x兲␸⬙共x兲 − 2⍀x␸⬘共x兲 = 0. The solution
obeying the required boundary condition at x = 1 is
␸bl共x兲 = const ⫻
III. GROUND-STATE CALCULATIONS
Throughout the rest of the paper we assume ⍀ 1. As E1
is exponentially small, the last term in Eq. 共6兲 is important
only in a narrow boundary layer near x = 1, and we can solve
Eq. 共6兲 for ␸1共x兲 ⬅ ␸共x兲 by using a matched asymptotic expansion; see e.g., Ref. 关34兴. In the “bulk” region −1 ⱕ x ⬍ 1
we can treat the last term in Eq. 共6兲 perturbatively. In the
zeroth order we put E1 = 0 and arrive at the steady-state equation 共1 + x兲␸⬙共x兲 − 2⍀x␸⬘共x兲 = 0, whose 共arbitrarily normalized兲 solution, bounded at x = −1, is ␸共0兲
b 共x兲 = 1. Now we put
␸b共x兲 = 1 + ␦␸b共x兲, where ␦␸b共x兲 1 and obtain in the first
order
关␦␸b⬘共x兲e−2⍀x共1 + x兲2⍀兴⬘ = −
2E1e−2⍀x
共1 + x兲2⍀ .
1 − x2
␸b共x兲 = 1 − 2E1
冕
0
e2⍀sds
共1 + s兲2⍀
冕
s
−1
␸b共x兲 ⯝ 1 − 2E1
冕
x
0
=1−
e2⍀sds
共1 + s兲2⍀
冉 冊冕
E1 e
⍀ 2⍀
2⍀
x
0
冕
共1 + r兲2⍀e−2⍀rdr
−1
e2⍀s
兵⌫关2⍀ + 1兴
共1 + s兲2⍀
− ⌫关2⍀ + 1,2⍀共1 + s兲兴其ds,
␸b共x兲 ⯝ 1 − 2E1
⬁
⌫共␣兲 − ⌫共␣,z兲 = 兺
j=0
共− 1兲 jz␣+j
,
j ! 共␣ + j兲
we can evaluate the integral in Eq. 共16兲 and obtain
共17兲
冕
x
0
⯝1−
⯝1−
共19兲
2E1冑␲
冑⍀
2E1冑␲
⍀3/2
e2⍀sds
共1 + s兲2⍀
冕
x
0
冕
⬁
共1 + r兲2⍀e−2⍀rdr
−1
2⍀s
e
ds
共1 + s兲2⍀
e2⍀x
.
共1 + x兲2⍀
共20兲
Now, by matching Eqs. 共19兲 and 共20兲, we obtain
E1 =
⍀3/2
2 冑␲
e−2⍀共1−ln 2兲
and
C = 1.
共21兲
One can see that the ground-state eigenvalue E1 is exponentially small in ⍀. Equation 共21兲 yields the mean extinction
time 共␮E1兲−1 共see Sec. V兲 which coincides with that obtained, by a different method, by Turner and Malek-Mansour
关18兴.
Equations 共15兲 and 共19兲 yield the ground-state eigenfunction
␸1共x兲 ⯝
共16兲
where ⌫共␣ , z兲 = 兰z⬁s␣−1e−sds is the incomplete gamma function 关35兴. Using the expansion 关36兴
册
e2⍀x
,
共1 + x兲2⍀
where C is a yet unknown constant. To find E1 and C we can
match the asymptotes of the bulk and the boundary-layer
solutions in the common region of their validity 1 / ⍀ 1
− x 1. Let us return to the first line of Eq. 共16兲 and evaluate
␸b共x兲 in this region. The inner integral receives the largest
contribution from the vicinity of r = 0, while the outer integral receives the largest contribution from the vicinity of s
= x. Therefore, we can extend the upper limit of the inner
integral to infinity and obtain 共see Appendix A兲
共1 + r兲2⍀e−2⍀r
dr. 共15兲
1 − r2
s
e2⍀s共1 + s兲−2⍀ds
1
冋
共14兲
This solution, which obeys the boundary condition 共7兲, is
almost constant in the entire region −1 ⱕ x ⬍ 1 except in the
boundary layer near x = 1 共to be defined later on兲. To find the
probabilities Pn共t兲, we will need to calculate the derivatives
of ␸b共x兲 at x = 0. As long as 1 − x 1 / ⍀, we can neglect the
r2 term in the denominator of the inner integral in Eq. 共15兲
and obtain
x
⯝ C 1 − e−2⍀共1−ln 2兲
Solving this equation, we obtain the bounded solution for
␸b共x兲
x
冕
再
␸b共x兲
␸bl共x兲
for 1 − x 1/⍀,
for 1 − x 1.
共22兲
Now we use Eq. 共11兲 to calculate the amplitude a1 entering
Eq. 共12兲. Let the initial number of particles be n0, so G共x , t
= 0兲 = xn0. Evaluating the integrals, we notice that 共i兲 the main
contributions come from the bulk region 1 − x 1 / ⍀ and 共ii兲
it suffices to take the eigenfunction ␸b共x兲 in the zeroth order:
n0
␸共0兲
b 共x兲 ⯝ 1. Furthermore, when n0 1, the term x under the
integral in the numerator is negligible compared to 1. So, for
n0 1, the numerator and denominator are approximately
031122-3
PHYSICAL REVIEW E 75, 031122 共2007兲
MICHAEL ASSAF AND BARUCH MEERSON
⬁
n̄共t兲 = 兺 nPn共t兲 = ⳵xG兩x=1 = ⍀e−␮E1t .
⬁
␴2共t兲 = n¯2 − n̄2 = 兺 n2 Pn共t兲 −
n=0
冉兺 冊
⬁
冋
nPn共t兲
3⍀
+ ⍀2共1 − e−␮E1t兲 e−␮E1t ,
2
共24兲
where we have used for ␸1共x兲 its boundary layer asymptote
␸bl共x兲, Eq. 共19兲, with C = 1. At intermediate times
⍀−1 ␮t E−1
1 one obtains a weakly fluctuating quasistationary 共metastable兲 state. Here the average number of particles,
n̄ ⯝ ⍀,
共25兲
coincides with the attracting point of the mean-field theory,
while the standard deviation
␴⯝
冉 冊
3⍀
2
1/2
P0共t兲 = G共x = 0,t兲 = 1 − e
−␮E1t
共27兲
which, at ␮E1t 1, is much less than unity. For n ⱖ 1 Eqs.
共3兲 and 共12兲 yield
冏
1 dn␸b共x兲
n! dxn
n(t)
1
冏
e −␮E1t ,
6
2 µE t 4
1
共28兲
6
0
0
2 µE t 4
6
1
FIG. 1. 共Color online兲 共a兲 The average number of particles as a
function of time at ␮t 1 / ⍀ as described by Eq. 共23兲 共solid line兲
compared with the prediction from the rate equation n̄共t兲 ⯝ ⍀
共dashed line兲, for ⍀ = 30. The inset shows a blowup at intermediate
times ⍀−1 ␮t E−1
1 where the curves almost coincide. 共b兲 The
standard deviation from Eq. 共24兲 versus time for the same ⍀.
Pn共t兲 =
2E1 共2⍀兲n−1e2⍀⌫共2⍀兲
−␮E1t
,
1F1共2⍀,n + 2⍀,− 2⍀兲e
⌫共2⍀ + n兲
n
共29兲
where 1F1共a , b , x兲 is the Kummer confluent hypergeometric
function 关35兴. To avoid excess of accuracy, we need to find
the large ⍀ asymptotics of Eq. 共29兲. To that aim we use the
identity 关35兴
共26兲
coincides with that obtained from the Fokker-Planck approximation; see Appendix B. Note that ␴共t兲 from Eq. 共24兲 is
a nonmonotonic function. This stems from the fact that the
quasistationary probability distribution around n ⯝ ⍀ decays
in time, whereas the extinction probability P0共t兲 grows. At
times ␮E1t 1 the standard deviation ␴ ⯝ 冑3⍀ / 2 corresponds to the unimodal quasistationary distribution around
n ⯝ ⍀, whereas at ␮E1t 1, ␴ → 0 corresponds to the unimodal Kronecker ␦ distribution at n = 0. At intermediate
times ␮E1t ⯝ 1, the distribution is distinctly bimodal. The
maximum standard deviation ␴max ⯝ ⍀ / 2 is obtained for
e−␮E1t ⯝ 1 / 2. Figure 1 shows the n̄共t兲and ␴共t兲 dependences.
Let us now proceed to calculating the complete probability distribution Pn共t兲 of the 共slowly decaying兲 quasistationary
state, conditional on nonextinction. For n = 0 we obtain
Pn共t兲 = −
µE t 0.05
10
2
n=0
册
0
0
0
0
2
= 兩关⳵xx
G + ⳵xG − 共⳵xG兲2兴兩x=1
=
12
共23兲
n=0
Furthermore,
30
20
What is the average number of particles n̄共t兲 and the standard deviation ␴共t兲 at times ␮t ⍀−1? Using Eqs. 共2兲 and
共12兲 with a1 = −1, we obtain
(b)
σ(t)
IV. STATISTICS OF THE QUASISTATIONARY STATE
AND ITS DECAY
30
18
(a)
n(t)
equal to each other up to a minus sign. Therefore, a1 ⯝ −1
共and independent of n0兲 which completes our solution 共12兲
for times ␮t ⍀−1.
1F1共2⍀,n
+ 2⍀,− 2⍀兲 =
⌫共n + 2⍀兲
⌫共2⍀兲⌫共n兲
⫻
冕
1
e−2⍀ss2⍀−1共1 − s兲n−1ds
共30兲
0
and consider separately two cases n 1 and n = O共1兲.
For n 1, the integral in Eq. 共30兲 can be evaluated by the
saddle point method 关34兴. Denoting ⌽共s兲 = −2⍀s + 2⍀ ln共s兲
+ n ln共1 − s兲 we obtain
Pn共t兲 ⯝
2E1 冑2␲共2⍀兲n−1e2⍀ −2⍀关s −ln共s 兲兴+n ln共1−s 兲−␮E t
*
*
*
1 ,
e
n! 冑2⍀共1 − s*兲2 + ns2*
共31兲
where s* = 1 + q − 共q2 + 2q兲1/2 is the solution of the saddle
point equation ⌽⬘共s兲 = 0 and q = n / 共4⍀兲. Equation 共31兲 can
be simplified in three limiting cases. In the high-n tail
n ⍀ 1, we have s* ⯝ 2⍀ / n 1 and
x=0
where ␸b共x兲 should be taken from Eq. 共16兲. After some algebra 共see Appendix C兲, we obtain, for n ⱖ 1,
031122-4
Pn共t兲 ⯝
冉 冊
22⍀−3/2 2⍀
冑␲ n n
n+2⍀
en−2⍀−6⍀
2/n−␮E t
1
.
共32兲
PHYSICAL REVIEW E 75, 031122 共2007兲
SPECTRAL THEORY OF METASTABILITY AND…
1.1
Enum
/E1
1
−10
n
ln P (t)
−5
−15
1
0.9
−20
−25
20
40
0.8
20
60
n
FIG. 2. 共Color online兲 The natural logarithms of the analytical
result 共31兲 for the quasistationary distribution 共dots兲, of the distribution obtained by a numerical solution of the 共truncated兲 master
equation 共1兲 共solid line兲, of the stationary solution 共B2兲 of the
Fokker-Planck equation 共dashed line兲, and of the Gaussian distribution 共34兲 共dash–dotted line兲, for ⍀ = 30 and ␮E1t 1.
In the low-n tail 1 n ⍀, we have s* ⯝ 1 − 冑n / 共2⍀兲 and
Pn共t兲 ⯝
冉 冊
22⍀−2 2⍀
冑␲ n n
n/2+1/2
en/2−2⍀−␮E1t .
共33兲
Finally, for 兩n − ⍀ 兩 ⍀, s* ⯝ 1 / 2 − 共n − ⍀兲 / 共6⍀兲, and we obtain
Pn共t兲 ⯝ 共3␲⍀兲−1/2e−共n − ⍀兲
2/共3⍀兲−␮E t
1
.
共34兲
For ␮E1t 1 this result describes a normal distribution with
mean ⍀ and variance 3⍀ / 2, in agreement with Eqs. 共25兲 and
共26兲 and with the predictions from the Fokker-Planck equation; see Appendix B.
Now we turn to the case of n = O共1兲. Then it is always
n ⍀. Here it is convenient to rewrite the integral in Eq.
共30兲 as
冕
1
e⌿共s兲s−1共1 − s兲n−1ds,
共35兲
0
where ⌿共s兲 = 2⍀共ln s − s兲. The function ⌿共s兲 has its maximum exactly at s = 1, the upper integration limit. The largest
contribution to the integral comes from the small region
O共1 / 冑⍀兲 near s = 1. Therefore, it suffices to expand ⌿共s兲 up
to the second order in 共s − 1兲2, replace the factor s−1 by 1, and
extend the lower integration limit to −⬁. The result is
e−2⍀
冕
1
2
e−⍀共s − 1兲 共1 − s兲n−1ds =
−⬁
e−2⍀ ⌫共n/2兲
.
2⍀n/2
60
80
µt
100
FIG. 3. 共Color online兲 Shown is the ratio of the numerical
ground-state eigenvalue Enum
from Eq. 共38兲 and the approximate
1
analytical value of E1 from Eq. 共21兲, for ⍀ = 20 and n0 = 100. The
deviation from 1 is about 5.6%—that is, within error O共1 / ⍀兲.
The Fokker-Planck approximation strongly underpopulates
the low-n tail and overpopulates the high-n tail. On the contrary, the Gaussian approximation strongly overpopulates the
low-n tail and underpopulates the high-n tail. Our analytical
solution 共31兲 is essentially indistinguishable from the numerical result, even at small values of n. Actually, it is in
good agreement with the numerics already for ⍀ = O共1兲, and
the agreement improves further as ⍀ increases.
We also computed the ground-state eigenvalue by solving
Eq. 共4兲 numerically with the boundary conditions G共1 , t兲 = 1,
⳵xG共−1 , t兲 = 0 and the initial condition G共x , t = 0兲 = xn0. At
times ␮t 1 / ⍀, the numerical ground-state eigenvalue Enum
1
can be found from the following expression:
Enum
=−
1
1
ln关1 − Gnum共0,t兲兴,
␮t
共38兲
where Gnum共x , t兲 is the numerical solution for G共x , t兲 and the
result in Eq. 共38兲 should be independent of time. A typical
example is shown in Fig. 3, and a good agreement with the
theoretical prediction 共21兲 is observed.
V. STATISTICS OF THE EXTINCTION TIMES
The quantity P0共t兲, given by Eq. 共27兲, is the probability of
extinction at time t. The extinction probability density is
p共t兲 = dP0共t兲 / dt. Using Eq. 共27兲, we obtain the exponential
distribution of the extinction times:
共36兲
p共t兲 ⯝ ␮E1e−␮E1t
at ␭t 1.
共39兲
The average time to extinction is, therefore,
Therefore, for n = O共1兲, we obtain
2E1共4⍀兲n/2−1⌫共n/2兲 −␮E t
e 1,
Pn共t兲 ⯝
n!
40
¯␶ =
共37兲
which, for n 1, coincides with that given by Eq. 共33兲.
Figure 2 compares our analytical result 共31兲 with 共i兲 a
numerical solution of the 共truncated兲 master equation 共1兲
with 共d / dt兲Pn共t兲 replaced by zeros and P0 = 0, 共ii兲 the prediction from the Fokker-Planck equation for this problem 关Eq.
共B2兲 of Appendix B兴, and 共iii兲 the Gaussian distribution 共34兲
for ␮E1t 1. In the central part all the distributions coincide.
冕
⬁
0
tp共t兲dt ⯝ 共␮E1兲−1 =
2冑␲ 2⍀共1−ln 2兲
e
.
␮⍀3/2
共40兲
This is in full agreement with the result of Turner and MalekMansour 关18兴 and in disagreement with the prediction from
the Fokker-Planck approximation, given by Eq. 共B5兲, and
with the prediction from the Gaussian approximation, given
by Eq. 共B7兲; see Appendix B.
Figure 4 compares the analytical result 共27兲 for P0共t兲 with
num
the extinction probability Pnum
共0 , t兲 found by solv0 共t兲 = G
031122-5
PHYSICAL REVIEW E 75, 031122 共2007兲
Extinction probability P0(t)
MICHAEL ASSAF AND BARUCH MEERSON
1
APPENDIX A
Here we will derive the result given by Eq. 共20兲. First, we
calculate the inner integral
0.5
I1 =
冕
⬁
共1 + r兲2⍀e−2⍀rdr =
−1
0
0
4
8
µt
冉 冊
e
2⍀
2⍀
⌫共2⍀兲.
共A1兲
As ⍀ 1, we can use the large-argument asymptotics of
the ⌫ function and obtain
4
12 x10
FIG. 4. 共Color online兲 Shown are the extinction probability
P0共t兲 关Eq. 共27兲兴 共dashed line兲 and the numerical solution of Eq. 共4兲
共solid line兲 at x = 0, for ⍀ = 20 and n0 = 100.
I1 ⯝
We calculated, at intermediate and long times, the complete probability distribution, including the quasistationary
distribution of the long-lived metastable state, and the extinction time statistics in a 共non-single-step兲 branchingannihilation reaction. To this end we employed the spectral
method, recently developed by the authors 关28,29兴. We also
used this example to illustrate how the “lacking” boundary
condition of the spectral method emerges in the theory.
The spectral method reduces the problem of finding the
statistics to that of finding the ground-state eigenvalue and
eigenfunction of a linear differential operator emerging from
the generating function formalism. The quasistationary distribution that we have calculated analytically is in excellent
agreement with numerics. The two widely used “rival”
approximations—the Fokker-Planck approximation and its
reduced version, the Gaussian approximation—perform well
only in the peak region of the quasistationary distribution.
They both fail in the tails of the distribution and, as a result,
cause exponentially large errors in the estimates of the mean
extinction time.
It is worth reiterating that, for single-step birth-death systems, the quasistationary distribution can be found directly
from a recursion equation for Pn, obtained by putting P0 = 0,
assuming a zero flux into the empty state, and replacing
共d / dt兲Pn共t兲 by zeros in Eq. 共1兲; see, e.g., 关12兴. For multi-step
systems such recursion equations are not generally soluble
analytically.
In conclusion, the spectral method is a powerful tool for
calculating the quasistationary distributions and extinction
time statistics of a host of multistep birth-death processes
possessing a metastable state and an absorbing state.
␲
.
⍀
Second, at ⍀ 1, the integral
I2 =
ing Eq. 共4兲 numerically as described at the end of the previous section, and a very good agreement is observed.
VI. FINAL COMMENTS
冑
冕
x
0
e2⍀s
ds ⬅
共1 + s兲2⍀
共A2兲
冕
x
e2⍀⌼共s兲ds,
where ⌼共s兲 = s − ln共1 + s兲, receives the largest contribution in
the vicinity of s = x 共remember that 1 − x 1兲. Then, expanding ⌼共s兲 around s = x, we obtain
⌼共s兲 = x − ln共1 + x兲 +
x
共s − x兲 + ¯ .
1+x
共A4兲
Extending the lower integration limit to −⬁ and evaluating
the remaining elementary integral we obtain, in the leading
order,
I2 ⯝
e2⍀x
.
⍀共1 + x兲2⍀
共A5兲
APPENDIX B
What are the predictions from the Fokker-Planck 共FP兲 approximation for the quasistationary distribution and the mean
extinction time of the branching-annihilation problem? The
FP description introduces an 共in general, uncontrolled兲 approximation into the exact master equation 共1兲 by assuming
n 1 and treating the discrete variable n as a continuum
variable. The FP equation can be obtained from Eq. 共1兲 by a
Kramers-Moyal “system size expansion” 关1,2兴 共in our case,
expansion in the small parameter ⍀−1 1兲. Using this prescription, we obtain after some algebra
再
⳵ P共n,t兲 ␮
⳵
=
− 关2n共⍀ − n兲P共n,t兲兴
⳵t
2
⳵n
+
冎
1 ⳵2
关2n共2n + ⍀兲P共n,t兲兴 .
2 ⳵ n2
共B1兲
The quasistationary distribution of the metastable state corresponds to the 共zero-flux兲 steady-state solution of the FP
equation. In the leading order in 1 / ⍀ we obtain
Pst共n兲 ⯝ 共3␲⍀兲−1/2e⍀−n+共3/2兲⍀ ln关共2n+⍀兲/共3⍀兲兴 ,
ACKNOWLEDGMENTS
We are very grateful to Alex Kamenev, Vlad Elgart, and
Lev Tsimring for useful discussions. Our work was supported by the Israel Science Foundation 共Grant No. 107/05兲
and by the German-Israel Foundation for Scientific Research
and Development 共Grant No. I-795-166.10/2003兲.
共A3兲
0
共B2兲
where only the central 共Gaussian兲 part of the distribution
contributes to the normalization. In fact, the distribution 共B2兲
is accurate only in the peak region 兩n − ⍀ 兩 ⍀ 共see Sec. IV兲,
where it reduces to a Gaussian distribution with mean ⍀ and
variance 3⍀ / 2:
031122-6
PHYSICAL REVIEW E 75, 031122 共2007兲
SPECTRAL THEORY OF METASTABILITY AND…
PGauss共n兲 = 共3␲⍀兲−1/2e−共n − ⍀兲
2/共3⍀兲
共B3兲
.
Were the FP equation 共B1兲 valid for all n, one could use it to
find the mean time to extinction ␶FP 共conditional on nonextinction prior to reaching the quasistationary state兲 by a standard calculation; see, e.g., Ref. 关1兴. This calculation would
yield
␶FP ⯝ 2
冕 冉
⍀
en 1 +
0
2n
⍀
冊
−3⍀/2
dn
冕
⬁
冉 冊
冑
共B5兲
Comparing it with Eq. 共40兲, one can see that the FP approximation gives a poor estimate of the mean extinction time, as
it introduces an exponentially large error.
The central 共Gaussian兲 part of the quasistationary distribution, 兩n − ⍀ 兩 ⍀, Eq. 共B3兲, can be correctly obtained
by keeping only leading-order terms, in the small parameter
兩n − ⍀ 兩 / ⍀ 1, in the FP equation:
再
冎
1 ⳵
⳵ P共n,t兲 ␮
⳵
=
− 关2⍀共⍀ − n兲P共n,t兲兴 +
关6⍀2 P共n,t兲兴 .
2 ⳵ n2
⳵t
2
⳵n
2
共B6兲
Indeed, the zero-flux steady-state solution of this equation
yields the Gaussian distribution 共B3兲.
Finally, what would be the prediction for the mean extinction time from the reduced FP description—that is, the one in
terms of Eq. 共B6兲? Here one would obtain
␶gauss ⯝ 2
冕
⍀
0
e
n2/共3⍀兲−2n/3
dn
冕
⬁
n
共C2兲
h共x兲 = e2⍀x .
As ⍀ 1, the inner integral receives its main contribution
from the vicinity of k = ⍀. The outer integral receives its
main contribution from the vicinity of n = 0. Therefore, one
can use the saddle point method for the inner integral and a
Taylor expansion for the outer one. The result is
␶FP ⯝
兵⌫关2⍀ + 1兴 − ⌫关2⍀ + 1,2⍀共1 + x兲兴其
,
共1 + x兲2⍀
f共x兲 =
dk.
共B4兲
␲ 1 ⍀关共3/2兲ln 3−1兴
e
.
3 ␮⍀3/2
e2⍀x
共1 + x兲2⍀
Let us introduce two auxiliary functions
e−k 1 +
n
2⍀
⫻ 兵⌫关2⍀ + 1兴 − ⌫关2⍀ + 1,2⍀共1 + x兲兴其. 共C1兲
3⍀/2
2k
⍀
␮k共2k + ⍀兲
冉 冊
E1 e
⍀ 2⍀
␸b⬘共x兲 = −
冑3␲ ⍀/3
e2k/3−k /共3⍀兲
dk ⯝
e ,
2
3␮⍀
␮⍀3/2
Using Eqs. 共C1兲 and 共C2兲, we can write the nth derivative of
␸b共x兲 关that is, the 共n − 1兲th derivative of ␸b⬘共x兲兴 at x = 0 as
冏
dn␸b共x兲
dxn
which again gives an exponentially large error as compared
with the accurate result 共40兲.
冉 冊
2⍀ n−1
共n − 1兲!
兺
k=0 k ! 共n − k − 1兲!
⫻ 兩 f 共k兲共x兲兩x=0兩h共n−1−k兲共x兲兩x=0 ,
共C3兲
where f 共k兲共x兲 is the kth derivative of f共x兲 and the same notation is used for h共x兲.
After some algebra, we find that the kth derivative
共k ⱖ 1兲 of f共x兲 at x = 0 is 关35,36兴
冏 冏
dk f共x兲
dxk
= 共− 1兲k2⍀关⌫共2⍀ + k兲 − ⌫共2⍀ + k,2⍀兲兴.
x=0
共C4兲
Now, the kth derivative of h共x兲 at x = 0 is
冏 冏
dkh共x兲
dxk
= 共2⍀兲k .
共C5兲
x=0
Using Eqs. 共28兲 and 共C3兲–共C5兲, we obtain for n ⱖ 1
Pn共t兲 = e−␮E1t
冉 冊
2E1 e
n 2⍀
2⍀ n−1
兺
k=0
共− 1兲k共2⍀兲n−k−1
k ! 共n − k − 1兲!
⫻ 关⌫共2⍀ + k兲 − ⌫共2⍀ + k,2⍀兲兴.
共C6兲
Actually, for n = 1 one has
P1共t兲 ⯝ 共␲/⍀兲1/2E1关1 + O共⍀−1/2兲兴,
2
共B7兲
冏
E1 e
=−
⍀ 2⍀
x=0
and the subleading term O共⍀−1/2兲 has been neglected in Eq.
共C6兲. Finally, using Eq. 共17兲 and changing the order of summation in Eq. 共C6兲, we obtain the following result for n ⱖ 1:
Pn共t兲 =
2E1 共2⍀兲n−1e2⍀⌫共2⍀兲
−␮E1t
,
1F1共2⍀,n + 2⍀,− 2⍀兲e
⌫共2⍀ + n兲
n
共C7兲
APPENDIX C
Here we calculate the nth derivative of ␸b共x兲, given by
Eq. 共16兲, at x = 0. The first derivative is
where 1F1共a , b , x兲 is the Kummer confluent hypergeometric
function 关35兴. To avoid excess of accuracy, we need to work
with the large-⍀ asymptotics of this result; see Sec. IV.
031122-7
PHYSICAL REVIEW E 75, 031122 共2007兲
MICHAEL ASSAF AND BARUCH MEERSON
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