-
--I
~d
f'iB
A.S. Mikhailov, Selectedtopicsin fluctuationalkineticsof reactions
condition D
321
is satisfied,the particle has a bound state in this potential, with an energylevel lying
nearthe bottom of the well. In this case,the maximal rate of growth of M2 is given by eq. (1.2.23). In
theopposite case(when D ~ uor~),the potentialwell is very shallowandthe dominantrole is playedby
the "kinetic energy" term in eq. (1.2.27). In a three-dimensionalmedium such a potential well has no
boundstatesand, consequently,the maximal rate of growth of the second-ordermoment is determined
by the lowest level of the continuous spectrum, i.e.
r
1
I
<{
uor~
Q2(D) = 2(uo - r),
d = 3.
(1.2.28)
,
.
This rate of growth does not depend on D. On the other hand, in a one-dimensional medium, a shallow
potential well always has a bound state. Its energy level lies close to the top of the well; it can be found
for an arbitrary form of a potential well [16]. By using this classicalresult of quantum mechanics,we
J
~
find
=
)
Q2(D) = 2(uo - r)
+ (2D)-I( J
II
(1.2.29)
two-dimensional medium, a shallow potential well also has a bound state, but its depth is
exponentiallysmall [16]. In this casewe find
In a
.~
,)
I
d = 1.
-=
,[
.
..
u(x) dx) 2,
Q2(D) = 2(uo - r)
+A
~exp[-2D(J
ro
=
u(r)r dr) -1],
d = 2.
(1.2.30)
0
-
I)
HereA is a dimensionless
numericalfactor.
'I
We see that, in the limit of fast diffusion, the behavior of the density-density correlation function
significantlydepends on the dimensionality d of the medium. For a three-dimensionalmedium, the
normalizedrate of growth! Q2(D) of this correlation function coincideswith that of the first statistical
IC -
1.\
moment.However,in the caseof d = 1 or d = 2, this normalizedrate of growthis still slightlylarger
than Ql = Uo - r. Hence, in low-dimensionalmedia, diffusion cannot prevent intermittency.This
1t
implies that the density-density
I~
correlation function might increase in time even below the formal
explosion threshold. On the other hand, in systems with high dimensionalities (d ~ 3) intermittency is
present only if the condition (1.;2.24) is satisfied, i.e. if diffusion is sufficiently weak.
1)
1.3. Explosions in media with randombreedingcenters
.n
Above we considered a special example of a system with random breeding and decay since it was
assumedthat random variations in breeding and decay rates are Gaussian and, furthermore, that they
tY
have vanishingly small correlation times. Now we want to discuss (see refs. [17-19]) a more general
7
~ituation. Namely, below we suppose that breeding occurs only within certain breeding centers that are
i)
Independently created at random locations at random time moments. AJI such centers are identical, i.e.
they have the same form, size and lifetime. In the limit when all breeding centers are very weak and
1L" largely overlap, the resulting breeding rate will be a Gaussian random field.
The principal problem is, what would be the critical frequency of such breeding regions (i.e. their
averagenumber per unit volume per unit time) that leads to explosion? To discuss it, we start with the
c,.=
~~-
--
.~
.,k
~~
;
cc=~
~
'"
~
1
322
A.S. Mikhailov,Selected
topicsin fluctuational
kineticsof reactions
simplestcaseof ideal mixing. If diffusion of reactingparticlesis extremelyfast, their distributionis
alwaysuniform and we can neglectall inhomogeneities.Then such a systemis describedby the
stochasticdifferential equation
n=-an+f(t)n,
(1.3.1)
wherea is the constantdecayrate andf(t) is the randomrate of breedingwhichrepresentsa certain
randomprocesswith givenstatisticalproperties.
The explosionthresholdis exceededif the mean density (n( t)), averagedover an ensembleof
randomrealizations,increaseswith time. The density(n(t)) canbe calculatedby direct integrationof
eq. (1.3.1):
t
(n(t)) = noe-at(exp(f f(t') dt')) .
(1.3.2)
0
Note that the last factor in eq. (1.3.2) can be expressedin termsof the generatingfunctionalof thc
randomprocessf( t),
t
= (exp(f f(t')u(t') dt')) .
<P[u(t)]
(1.3.3)
0
Comparisonof (1.3.2) and (1.3.3) showsthat
(n(t)) = noe-at<P[1],
(1.3.4)
where <P[1]is the valueof the functional (1.3.3) for u(t) ==1.
Equation(1.3.4) allowsus to determinethe explosionthresholdfor an arbitraryrandomlyvarying
rate of breedingf(t). Supposeit consistsof randomindependentpulsesof durationToand intensityJ,
so that
,
f(t) = J()(t- tj) ,
(1.3.5)
L
j
where ()(t) = 1 for 0 < t < To and ()(t) = 0 for t < 0 and t> To'
The generatingfunctionalof the Poissonrandomprocess(1.3.5) hasthe form
t
<P[u(t)]
t1
= exp{-mf [ 1- exp(J f ()(tl- tz)u(tz)dtz)] dtl} ,
0
(1.3.6)
0
wherem is the meannumberof pulsesper unit time. Substituting
~!t) ==1 into eq. (1.3.6),we find
<P[l]= exp{-m[1- exp(JTo)]}
(1.3.7)
and hence,accordingto eq. (1.3.4),
~"=~~~""
~
I
.,
,-
A.S. Mikhailov,Selected
topicsin fluctuational
kineticsof reactions
(n(t))
= no exp{[m(exp(JT
323
0) -1) - a]t} .
(1.3.8)
Therefore,the explosion threshold is reached when the mean number of pulsesper unit time exceeds
the critical value
mcr
= a[exp(JT 0) -1]-1 .
(1.3.9)
When JT 0 ~ 1, the breeding pulsesare weak and eq. (1.3.9) reducesto mcr= alJT o. This result can
beeasily obtained by equating the mean breeding rate (f) = mJT0 to the decay rate a.
On the other hand, if JT o}> 1 and the breeding pulses are strong, we have mcr= a exp(JT0)' This
correspondsto equating the decay rate a to the product of the effective breeding factor exp(JT0) for a
singlepulse and the mean number of such pulses per unit time m.
Hence, we seethat under the condition of an ideal mixing there is a rigorous solution to the problem
of finding the explosion threshold. This problem becomes, however, much more complicated if
diffusion is not infinitely fast and the effects of spatial distribution should be taken into account. The
correspondingmathematical model is
ri = -an + f(r, t)n + DJ1n,
(1.3.10)
wheren is now the population density of the breeding particles and D is their diffusion constant.
We assume that breeding occurs only within randomly created breeding centers, so that the
fluctuatingfield f(r, t) is given by a sum of random pulses,
f(r, t) =
L g(r -
rj' t - tj) .
(1.3.11)
j
All breeding centers are identical; any single center is described by the function g(r, f),
g(r, t) = J</>(r){}(t).
(1.3.12)
,,
Here {}(t) = 1 for 0 < t < To and (}(t) = 0 for t < 0 <Jr t> To; the function </>(r)falls rapidly to zero for
r}> ro and </>(0) = 1. Therefore an individual breeding center has the intensity J, the spatial size ro, and
thelifetime To' We assumethat the mean number of breeding centersper unit time per unit volume is
constantand equal to m. Below we shall frequently use the dimensionlessspace-time concentrationc of
breedingcenters,
c = mrgT0'
(1.3.13)
whered is the dimensionality of the medium. Note that in the limit c}> 1 the fluctuating field f(r, t)
becomesGaussian.
All breeding centers may be divided into strong and weak, depending..onthe relative increaseof the
populationdensity on an individual center.
The increasein population density at an individual breeding center is describedby an equation
-ri=-DJ1n-J</>(r)n,
;;~=::",=cc=o,--=;;
~"
~
"
,7"'"';C~=="""--co=c--,",,=~=-=C'-7,,,,-~=~:-c
O~t~To'
(1.3.14)
c-~
-.
~~=-
-,-
~--
ii,;
"
~-
~
324
A.S. .Wikhailov,Selectedtopicsin fluctuationalkineticsof reactions
We have multiplied both its sides by -1 so that it becomes formally identical with the Schrodinge
equation with imaginary time and the potential U = -J~(r). Its general solution is
r
n(r, t) =
~
~Cjl/lj(r)exp(njt) + J C(n)l/In(r) entdn ,
(1.3.15)
J
where the sum is taken over the discrete spectrum and the integral is taken over the continuous
spectrum of the linear operator
..
L=DL1+J~(r).
(1.3.16)
For breeding centers we have J> 0 and therefore the eigenvaluesnj in the discrete spectrum arl.'
positive. They correspondto negative energy levels of bound statesin the potential well U(r) = -J(p(r).
Supposethat no is the largest of the eigenvaluesin the discrete spectrum. If no To~ 1, the breeding
centersare strong, since any such center already leads to exponentiallylarge production of thl.'
population. If the opposite condition noT 0 <{ 1 is satisfied, the breeding centers are weak. We also S.IY
that a breeding center is weak if the linear operator L has no discrete spectrum [there are no bound
states in the corresponding potential well U(r)].
The maximal eigenvalue no can be estimated (see ref. [18]) by using the analogy with thl.'
I
Schrodingerequationand recallingthat no correspondsto the lowest-lyinglevel in the potentialWI.'II
U(r) = -J~(r). This allowsus to approximatelyexpressno in termsof the intensityJ and the spati.tl
size roof a breeding center, for different dimensionalities of the medium. Then the condition no To~ I
that definesstrongcenterscanbe rewrittenasJ ~ J*. Therebywe introduce the characteristicintensity
J* which allows us to distinguish between strong and weak breeding centers. It can be shown [18] that
for the short-lived ( To<{ r~/D) centers we have
J* = T~l,
d = 1,2,3.
(1.3.17)
If the breeding centers are long-lived ( To <{ r~/D), the value of J* dependson the dimensionality d of
the medium,
\
J* = (D/r~To)1/2,
d = 1,
J* = (D/r~)[ln(DTo/r~)J-1,
J* = D/r~,
(1.3.18.1
d = 2,
(1.3.18b)
d=3.
(1.3.1Rc)
Now, after these preliminary notes, we can return tq our principal problem which consistsin
determining the explosion threshold in such media. Below we construct the solution separatelyfor the
strong and the weak centers.
The simplest way to calculate the explosion threshold for strong centers is within a one-center
approximation.
W
-"
'j~~
In
this
approximation
we
neglect
the
interference
.,
between
.
different
centers.
To
obtain
an estimate for the explosion threshold we evaluate first the total increaseof population dNI on a single
center, then multiply it by the space-time concentration of breeding regions m and equatethe result to
the amount of the population that decaysper unit time per unit volume.
-
r_;~~
c'cc",
v
~~
~
-
-
--- -~~
i
'f
A.S. Mikhailov, Selectedtopicsin fluctuationalkineticsof reactions
325
The total increaseof the population at an individual strong center during its life is
~N1 = norf exp(floT 0) ,
(1.3.19)
where no is the initial population density and r 1 is the localization radius of the eigenfunction <l>o(r)of
the operator L corresponding to its maximum positive eigenvalue
r1
=(J l/Io(r)dr)z/d,
J l/Io(r)zdr=
1.
(1.3.20)
Thislengthcan be estimatedas r1~ ro for J~ Dlr~ and rl
(Dlflo)l/z otherwise.
Within the one-center approximation, as was noted above, we have the equation
~
m ~N1 = ano ,
(1.3.21)
sothat the threshold concentration of breeding centers is given by a simple expression
c~~)= aT o(r olr l)d exp( -floT 0)
.
(1.3.22)
To determine the limits of validity of the one-center approximation, we can evaluate the two-center
correctionsto the explosion threshold.
Supposethat the second center is created in a region where the population density is higher than
average,i.e. in a spreading spot of the additional population produced by the first center. Then,
breeding at the second center starts from the higher density and this increasesthe amount of the
producedpopulation. The total increaseof the population by a pair of breeding centerscan be written
as
~N
= ~N1 + ~Nz + ~N1z .
(1.3.23)
If thesetwo centershad appearedat poirits r 1 and r Z at times t 1 and tz, the additionalincreasein
population is
'
~N1z(rz- rl' tz - tl) = norf exp(2floT 0)
x J l/Io(r
- rz)l/Io(r' - rl)Gd(r - r', tz - tl - To) dr dr' ,
(1.3.24)
where G d is the Green function for the diffusion problem in d dimensions.
The average contribution resulting from additional production of the population by pairs of breeding
centersper unit time per unit volume is
(~N1z) = !m
J ~N1z(R,
T) p(R, T) dR dT,
-
...
(1.3.25)
where p(R, T) is the probability density that the secondcenter will be created at a distanceR after time
T following the appearance of the first. For the centers that are randomly and independently created,
such a distribution has the form
~~
.,,1
~~. =~
326
A.S. Mikhailov, Selectedtopicsin fluctuationalkineticsof reactions
p(R, T)
= m exp(-mVdT),
(1.3.26)
where Vd is the volume of a sphere of a radius R in a d-dimensional space.
After simple calculations (see ref. [18]) we find
(LlN1z) ::= no'idm exp(noT0) (m/ D)d/(d+Z).
(1.3.27)
The additional production of the population due to interference of the breeding centers leads to
,
lowering of the explosion threshold as comparedto the prediction of the simplestone-centertheory. By
using eg. (1.3.27) we obtain the corrected value for the explosion threshold,
ccr= c~~)[I- 'Yd(a/a*)d/(d+Z)],
(1.3.28)
where
a*
= (D/,i)
exp[ -(2/d)noT 0]
(1.3.29)
and Yd is a numerical factor that depends only on the dimensionality of the medium,
'Yd
= ir(i),
d = 1,
'Yd
= iv-.n:, d = 2,
'Yd= ~r(~)(3"i2/5)Zi5,
d = 3.
(1.3.30)
Here r(x) is the gamma function.
Since eg. (1.3.28) takes into account only the correction arising from pairs of breeding centersand
does not include additional contributions from larger clustersof breeding centers,we can expect that it
would hold only when this correction is small, so that even the most probable two-centerclustersdo not
influence significantly the prediction of the one-center theory. As follows from eg. (1.3.28), this
happensif a ~ a *. We expect that in the opposite limit a ~ a * contributions from the clusters of all
possiblesizesare of the sameorder of magnitude and the explosion threshold is significantly lower than
(0)
ccr
.
A similar estimate might be obtained by using a different line of arguments. It is"'clear that the
mutual influence of breeding centers can be neglected if the volume v of the space-time region
occupied by an enhanced-densityspot due to an individual center is much smaller than the average
space-time volume per siQglecenter
mv ~ 1 .
(1.3.31)
Let us estimate the volume v. In order to do this we consider first the density at the central point of
the spot as a function of time. When a breeding center disappears it leaves in the medium the
enhanced-densityspot of linear size about, 1 with a maximal density increment (at the central point) of
about on ::= no exp(no To). After the breeding center ceasesto operate, this spot begins to spreadout
due to diffusion, so that the additional density at the central point decreaseswith time as.
d
d
'
~~C~'~~-~_"""6
~
~
~
(1.3.32)
~~~.
-~---~
.~
'lon(O) ::= no (4;Dt)d72
'1
on(t)::= (4;Dt)'d72
exp(noT 0) .
-
-
~
;:~~;
1
,.-
-
~
i
A.S.Mikhailov,
Selected
topicsinfluctuational
kinetics
of reactions
)
327
The enhanced-density spot remains distinguishable from the mean background while 8n(t) ~ no'
Thereforewe can estimate the lifetime T* of a single enhanced-densityspot from the condition that
on(T*)::= no as
T*::= (Dlr~) exp(2floT old) .
)
(1.3.33)
Sincethe spot spreads diffusionally, its maximal linear size by the time moment T* is (DT*)1/2. Hence,
we can estimate the space-time volume v occupied by a single enhanced-densityspot as v::=
)
y
i.i
T*(DT*)d/2.
If we substitutenow this estimatefor
v into mv ~ 1, taking into accounteq. (1.3.33), we can derive a
condition for the mean space-time concentration m of the breeding centers which allows to neglect
their mutual influence. This condition reads
.
)
PI
m ~ (Dlr~+d) exp[-(1 + 2ld)floT 0] .
(1.3.34)
:i) If we require now that the threshold concentrationm~~)= rgToc~~)found within the one-center
I
approximationshould satisfy the condition (1.3.34), it can be easily verified that the resulting inequality
wouldbe simply the condition a ~ a * which was derived above by direct estimation of the corrections
1
dueto two-center clusters.
To summarizeour discussionof the explosion threshold for strong breeding centerswe can say that
thenaive one-centerestimate (1.3.22) for the threshold space-time concentration of breeding centers
remainsvalid [with small corrections given by eq. (1.3.28)] only provided that this threshold density
satisfiesthe condition (1.3.34) or, equivalently, if a ~ a* where a* is given by eq. (1.3.29). If this
conditionis violated (i.e. if a ~ a *) the explosion threshold is significantly lower than the one-center
predictionand it is then determined by the clusters of breeding centers.
Whenthe breedingcentersare weak, we can take, as the first approximationfor the explosion
threshold,the condition that the mean breeding rate (f) averaged over the entire volume of the
.I
t
.
~
I
~ mediumshouldbe equalto the decayrate a. Sincethe meanbreedingrate is
~
(f) = m J g(r, t) dr dt = clal "
'(1.3.35)
where
~
al
= r~dJ 4>(r)dr,
(1.3.36)
this gives a mean-field estimate for the dimensionless critical concentration of breeding centers
I
c~~)
= alaI I.
(1.3.37)
Now we want to evaluate the fluctuational corrections to this simpl.~result. As follows from eq.
(1.3.10),the mean population density (n) averagedover the entire volume of the medium changesin
timeaccordingto the equation
d(n) Idt
=-_"7=-==
= -(a - (f))(n)
CO~C-.
+ (8f 8n) ,
,=c~=,,--~=,"
(1.3.38)
~~-
---
328
A.S. Mikhailov, Selectedtopicsin fluctuationalkineticsof reactions
where &f = f - (f) and &n = n - (n). We seethat the mean-fieldpredictionfor the explosionthreshold
is valid only if correlations between density and breeding-rate fluctuations (i.e. &n and &f) can be
neglected.
Density fluctuations &n satisfy the equation
&1i
= -(a - (f))
&n
+ DL1 &n + &f(r, t)(n) + (&f &n- (&f &n)).
(1.3.39)
Since, near the threshold, the mean density (n) increasesor decreaseswith time very slowly in
comparison with all other characteristic times, it can be regarded as constant in eq. (1.3.39).
To calculate the fluctuational corrections to the explosion threshold we must use eq. (1.3.39) to
determine the correlation function (&f &n) at fixed (n). This can be done by applying the diagrammatic
perturbation technique [20-22] for stochastic differential equations. The detailed derivation can bL'
found in ref. [17]; below we indicate only the principal points of this analysis.
After the Fourier transformation with respect to time, eq. (1.3.39) takes the form
&nq
where Q
= G~((n) &fq + J (&fq-q'
=(n,
k) and G~ = (-in
&nq_q' - (&fq~q' &nq')) dQ') ,
(1.3.40)
+ a - (f) + Dk2)-\
The formal solution to the integral equation (1.3.40) is given by an infinite iteration series in powers
of &fq. Multiplying this infinite series by &fq' and averaging over the ensemble of fluctuations &f, WL'
obtain the expression for the correlation function (&nq &fq,) and, after integration, for the quantity
(&f(r, t) &n(r, f)). Graphically, the first few terms of the infinite series for this correlation function arc
~X~
&:.:~.:~ +
(&f(r, t) &n(r, f)) = (n)(
'.-
~
-0-
,',,'
"
"
'-;.0\
+
"
-G
-
-'
~
~,'
e
:..
+
-
;..
+
",
"
'
-;--0..,-,
'.1
.~
)
,
+...
-'
.
(1.3.41)
Here a thin solid line with an arrow representsthe function G~ whereasbroken lines with points upon
them represent irreducible correlators (cumulaats) of random fields &.1:
,. In contrast to the Gaussian
random fields, for which all higher-order correlators degenerateto p:oducts of pair correlators, thc
Poissonrandom field f(r, t) has nonzero cumulants of all orders. Therefore, several broken lines can
convergeto the samepoint.
;
Now
. we can introduce the full Green function Gq and the function I
q
related by the standardDyson
equatIon
G;l
= (G~)-l
- Iq .
(1.3.42)
By using Gq to perform a partial summation of the diagramsin the seriesfor Iq, we can write Iq in the
form
"-4-,,
Iq=
t-'~-.
--~'i'-"
+ f=*=~
...
,,' ,-,--0,-,
,
".4--,..,-0-,
+ ~~.~~
~-
+ ~~~~
,
"
+"'.
(1.3.43)
Here thick solid lines with arrows denote full Green functions Gq'
o-'+~-~
'"'~'"
~
1
.,~"('
(
A.S. Mikhailov, Selected
topicsin fluctuationalkineticsof reactions
329
Sincethe fluctuation correlator (8f(r, t) 8n(r, f)) can be expressedin terms of !q as (8f 8n) =
(n)1'o' the explosionthresholdis determinedby the equation
d
e
a = (f) + 1'0 .
(1.3.44)
Thus,!o representsthe fluctuationalcorrectionto the effectivebreedingrate. Note alsothat by solving
)
theDysonequationfor the full GreenfunctionGq we find
r1
G;l
= -in + a - (f) + Dk2 - 1'q.
(1.3.45)
1.;
Therefore,at the thresholdof explosion,this function hasa pole at q = O.
Now we can analyze contributions to 1'0 from different typesof diagrams,taking into accountthat
breedingcentersare weak and, consequently,we have a smallparameterI /I* <{1.
)
Such analysisleads to the following results. All diagramswith irreducible correlators of order higher
than 2 [e.g. the second and the fourth in (1.3.43)] are always small provided that I/I* <{ 1; the most
"dangerous" diagrams are those in which the broken lines cross [such as the third in (1.3.43)]. Their
contribution is not divergent and remains small as compared with that of the first diagram only if the
~
~
condition
c(ITo)(I/I*)
\
.
v
v
~
<{
1
(1.3.46)
issatisfied.Thereforethe mean-fieldexpression(1.3.37)only givesa correctestimatefor the explosion
thresholdprovided that the inequality (1.3.46) is still valid for the thresholdconcentrationc~~)of
breedingcenters.Substitutionof c~~)into (1.3.46)gives
aTo(I/I*)<{l.
(1.3.47)
Hence,weseethattheweakness
of breedingcenters(i.e. theinequalityI/I* <{1) doesnot guarantee
)
that the mean-fieldapproximationis applicable.To ensurethis, a much more stringentcondition
(1.3.47)shouldbe satisfied.This additionalconditionrequiresthat the lifetimes Toof weakpreeding
centersshould not be very long. When the condition (1.3.47) is violated, the contributionsof all
diagramswith crossingbroken lines are of the sameorder of magnitudeas that of the first diagram.
Thereforewe canexpectthat in this situationa substantialloweringof the explosionthresholdwouldbe
observed,in comparisonto the mean-fieldformal prediction.
The subsequent
analysisrevealsthat this sharpreductionin the explosionthresholdcanbe explained
by a specialrole playedby large, but rare clusters.In effect, althoughany suchclusterconsistsonly of
weakbreedingcenters,it can behavelike an individualstrongbreedingcenter.
Let us supposethat the volumeV containsan isolatedclusterof weakbreedingcentersduringa time
intervalt. The increasein the amountof the populationwithin this volumeis then describedby
t
,\
;:
1
I
I
.
Ii = f(r, t)n + Diln ,
,
~
~
(1.3.48)
...
wheref(r, t) is the breedingrate field for the chosencluster.
As was alreadynoted, this equationcan be interpretedas the Schr6dingerequationfor imaginary
timewith the potential U(r, t) = - f(r, f). Sincef(r, t) is a breedingrate and, therefore,it is positive,
""
,"
~-
--
'~C-~="'"
"-CO
7-~'-7--C-CC-C
===o~c~-=-;,~;;;c_,~.,;
-=cc===c
-~
330
A.S. Mikhailov, Selectedtopicsin fluctuationalkineticsof reactions
any such cluster correspondsto a certain time-dependentpotential well. For each given instant of time
we can calculate the spectrum of energiesEj(t) for bound statesin this potential well. These energie~
define the positive eigenvalues nj(t)
= -Ej(t)
of the linear operator
L=DL1+f(r,t).
(1.3.49)
The most rapidly growing contribution to the density n(r, t) comes from the term with the largest
positive eigenvalue n(t)=max{nj(t)}.
A cluster can be crudely characterizedby its time-averagedmaximal eigenvaluen and its lifetime T.
When nT ~ 1, such a cluster remains weak. On the other hand, when nT ~ 1, the increaseof thc
population on this cluster is exponentially large. Despite the fact that it consistsonly of weak breeding
centers, such a cluster behavesthen like a single strong breeding center. Obviously, this is possibleonly
if the cluster representsa very large aggregation of individual weak centers.
Note that the condition n T ~ 1 can be written also in the form T ~ n -1. Therefore, the potential
U(r, t) will be a slowly varying function of time as compared with the characteristictime scalen-1 of
motion inside such a potential well. Hence, the adiabatic approximation is valid and the total
production of the population by a strong cluster can be estimated as
IlN
= no exp(J
n(t) dt) .
(1.3.50)
We see that the total production of population by an individual strong cluster can be uniqucly
describedby a single parameter, the intensity of a given cluster, which is defined as
s = exp(J n(t) dt) .
(1.3.51)
Supposethat p(s) is the probability per unit volume per unit time to find a strong cluster with the
intensity s. The averageincreaseof population density per unit time due to strong clusters is then
(Iln)
= J IlN(s)p(s)
ds "
(1.3.52)
so that the effective breeding (ate due to strong clusters is
Q = J eSp(s)ds .
(1.3.53)
Evidently, this effective breeding rate can be small when compared to the averagebreeding rate (f)
exp[- c1>(s)]wherec1>(s)~ 1
only providedthat the strongclustersare exponentiallyrare,'i.e. if p(s)
-
for s ~ 1.
However, the exponential rarity of strong clustersis only a necessary,but not a sufficient, condition.
When such a condition is satisfied, we have
...
Q
- J exp[s-
c1>(s)]ds. .
(1.3.54)
-.c
-..,8..
~
.
,.
"
A.S. Mikhailov,Selected
topicsinjluctuationalkineticsof reactions
331
Hence, the order of magnitude of Q is determined by competition betweentwo factors: although strong
clustersare exponentially rare they are characterizedby an exponentially large production of population at any single cluster. Therefore strong clusters can give the dominant contribution to the overall
breeding rate and determine the explosion threshold even if they are very rare.
Note that the condition II T ~ 1, specifying a strong cluster, can be realized, principally, in two
different ways. First, it is possible that we have a chain of individual breeding centers that are
subsequentlycreated one after another within the samesmall volume of the medium. Sincein this case
there is mainly an overlap between the subsequentcenters or the enhanced-densityspots produced by
them, such aggregationsmight be called the time clusters. In the opposite limiting case of a space
clusterwe have a spatial aggregationof many individual breeding centers;the lifetime of a spacecluster
would be usually of the sameorder of magnitude as that of an individual breeding center. Hence, for a
typical time cluster we expect II - llo and T ~ To, whereas a space cluster typically has II ~ llo but
T- To'
Although all further estimatescan be carried out in the general caseof the Poissondistribution of
breedingcenters, for simplicity we restrict our discussionbelow to a Gaussianlimit which corresponds
to large space-time concentrationsc ~ 1 of the breeding centers. In this limit breeding centersare very
weak and largely overlap in spaceand in time, so that f(r, t) representsa Gaussianrandom field. It is
convenient to define its fluctuating component 8f as 8f(r, t) =f(r, t) - (f(r, f)). Then 8f(r, t) is a
Gaussiannoise with averageintensity S cJ2, correlation radius r 0 and correlation time To.
Since, in the Gaussianlimit, the breeding centers largely overlap, their strong "clusters" represent
now the space-time regions with anomalously high concentrations of breeding centers or, in other
words, the extremely strong rare positive bursts of the random field 8f(r, f). We also have to modify our
definition of time and spaceclusters. Let ll* be the characteristicdepth of the lowest-lying level in the
potential well that corresponds to a typical field fluctuation 8f(r, f). Since any such fluctuation lives
about time To, the correlation time of this random process,all typical fluctuations will be weak if the
condition II * To~ 1 is satisfied.
Strong fluctuations are characterizedby large values of the parameter s = II T. Principally,this can
be achievedin two different ways. Firstly, some of the fluctuations with the typical value ll-ll*
may
turn out to be unusually long-lived so that their lifetime T is much greater than To. Secondly, very
strong bursts for which II ~ llo may appear and, since they are very rare events, their lifeti~-e will be
most probably about To.
'
Below we estimate separately contributions to the effective breeding rate from the two principal
classesof strong fluctuations.*) It turns out that the long-living typical fluctuations (that can be viewed
as "time-clusters") are always insignificant and that the most important are the "space clusters", i.e.
very strong bursts of the breeding rate field with typical durations. It is preciselythese "space clusters"
that are responsiblefor sharp reductions in the explosion threshold.
For a Gaussianrandom process,the probability of appearanceof long-living fluctuationswith T ~ To
and II - llo can be estimated as
-
"'
p(T)
- exp[- cp(T)] ,
(1.3.55)
...
where
0>Certainly, there are also the intermediate cases. A general method, allowing one to estimate contributions of all strong rare fluctuations, is
provided by the technique of an optimal fluctuation (see appendix B). However, since we are interested here only in crude estimates for the
explosion threshold, such a more sophisticated analysis does not lead to any significant corrections.
~~
~~
~
~~c.c-~-7~7-=
=;cc;;c=~~';;;;-=~i'C"=~=;j;",:"
~
~:;:~~,~-=-~,~,c=-c
---
332
A.S. ,Wikhailov,Selectedtopicsin fluctuationalkineticsof reactions
cp(T)
- TIT
0'
for T ~ To'
(1.3.56)
Hence, their contribution to the effective breeding rate is about
~
Q
-J
exp[{l* T - cp(T)] dT
(1.3.57)
lID'
and, since {l* To~ 1, it is always exponentially small and can be safely neglected.
To estimate the probability of strong "space clusters" which are characterizedby the conditions
{l ~ {lo and T - To we can use certain results from the theory of the Schrodinger equation with a
stationary random potential. Indeed, within the time intervals about To the fluctuating field [(r, t)
remains almost stationary becauseTo is the correlation time for this random field. Let us recall now that
U = -[(r, t) plays the role of a potential in eq. (1.3.48) that can be interpreted as a Schrodinger
equation with imaginary time. We see that this random Gaussian potential remains approximately
stationary within a lifetime of a single strong "space cluster". Therefore we can estimatethe probability
of finding a cluster with the largest positive eigenvalue {l from the probability distribution for the
energy levels E = - {l in a random stationary Gaussianpotential in the so-called"fluctuational" part of
the energy spectrum where the density of levels is already exponentially small. By using the formulas
for the density of energy levels in this part of the spectrum, that were derived in refs. [23, 24], and
changing the relevant notations, we find
- exp[- $({l)]
p({l)
,
(1.3.58)
where $( {l) is given by the expressions
= a{l2-(dI2)DdI2S-lr~d
$({l)
,
(1.3.59)
when {lr~1D ~ 1, and
$({l)
= {l2/2S "
(1.3.60)
where {lr~ ~ 1.
Equations (1.3.59) and (1.3:60) hold while $({l) ~ 1. The numerical factor a in eq. (1.3.59) is of the
order of unity.
The contribution from the "space clusters" to the effective breeding rate can hencebe estimatedas
~
Q
-J
exp[F({l)] d{l ,
T-l0
(1.3.61)
.,.
where the function F( {l) is defined as
F({l)
=;;=-"",
= {IT
0
- $({l)
.
(1.3.62)
-0;
'°,,5\;'.
~
""f
.
A.S. Mikhailov, Selectedtopicsin fluctuationalkineticsof reactions
333
The integral in eq. (1.3.61) should be evaluatedseparatelyfor the media of different dimensionalities
and for the situations when the breeding centers have long or short lifetimes. Let us recall that the
lifetime To is said to be long if To~ r~/D and short in the opposite case.
In one-dimensionalmedia with long-lived breeding centers such estimation showsthat the contribution from strong fluctuations to the effective breeding rate and, hence, to the explosion threshold is
alwaysnegligibly small provided that all strong fluctuations with s?1 (or n? T;l) are exponentially
rare. By using eq. (1.3.59) with d = 1, we find that this condition is satisfied if
.
S ~ (D/T~r~)1/2 .
By taking then into account that S
- cJ2 and by using an estimate (1.3.18a) for J*, we obtain
(cJ) To(J/J*) ~ 1 .
~
(1.3.63)
(1.3.64)
Note that this inequality is precisely the same as the condition (1.3.46) derived above by direct
evaluation of various terms of the perturbation expansion. If we substitute into (1.3.64) the threshold
concentration c~~)in the mean-field approximation, we can obtain again condition (1.3.47) which
guaranteessmallnessof the fluctuational lowering of the explosion threshold and justifies the mean-field
approximation. We see now that it is simply the condition that the strong fluctuations should be
exponentially rare at the explosion threshold. The probability of these strong fluctuations dependson
the space-time concentration c of the breeding centers at the explosion threshold. By increasing the
decay rate a we increase the threshold concentration and finally come to a situation where strong
fluctuations, belonging to the classof "space clusters", ceaseto be exponentially rare and they start to
give the dominant contribution to the effective breeding rate, sharply reducing the explosion threshold.
We can expect that this would happen when a will reach the value, at which the condition (1.3.47) is
violated. However, we are not able to give any quantitative estimatesfor the explosion threshold in this
region becauseno explicit formulas for the probability of strong fluctuations are available if such
fluctuations are not exponentially rare. Here we encounter, in effect, a classical and extremely
complicatedproblem of finding the density of energy levels in the intermediate region, between the
fluctuational and the typical parts of the energy spectrum.
,\
Analysis of the two-dimensional case with long-living centers leads to analogousconclusions.The
only difference is that, when this analysisis performed, it does not permit us to obtain the logarithmic
factor which is present in the expression(1.3.18b) for J* in this case. However, this can be expected
becausethe above estimatesdo not take into account the preexponentialfactor in the expressionfor the
probability distribution p( n).
The three-dimensional model with long-living breeding centers presents a special case. Here the
violation of the condition (1.3.46) is again connectedwith the sharp increasein the contribution from
rare strong fluctuations of the "space cluster" type. Howe,ver, in contrast to the one- and twodimensionalcases,this sharp increase can now take place even when all strong fluctuations are still
exponentiallyrare. When S increases,a maximum in the dependenceof Fon n developsat n D/r~.
It starts to grow and, at a certain critical value of S, the curve touches..thefl-axis. For higher valuesof
the intensity S there exists an interval of fl, in the vicinity of the point n = D / r~, where the function
F(n) is positive and it is greater than unity so that it gives a very large contribution to the effective
breeding rate Q. Then the strong fluctuations with fl ~ D/r~ start to playa dominant role and the
~
mean-fieldapproximation becomesinapplicable.
~~~~;.;;~
~
-
--~
334
A.S. Mikhailov,Selected
topicsin fluctuational
kineticsof reactions
The critical intensity Scrat which this happenscan be estimated, by using eqs. (1.3.59), (1.3.60) and
(1.3.62), as Scr-- D/T or~. For smaller intensities S ~ Scr the mean-field approximation should hold. If
we recall now that S -- cJ2 and that in this caseJ* = D/r~, we immediately can see that the condition
S ~ Scrcoincides again with the inequality (1.3.46) found by estimation of the terms in the perturbation
expansIon.
In the mean-field approximation the threshold concentration of the breeding centers is c~~)-- all.
When the decay rate a increases,this threshold concentration also grows until it reachesthe values
wherethe strongfluctuationswith n ~ D/r~ start to play the dominant role. This occurs at a :::= a =
J*/JTo' For higher values of the decay rate, within the interval (DTo/r~)3/2~ln(a/ac)~DTo/r~, the
explosion threshold can be found by using a saddle-point approximation for the integral (1.3.61)
ccr = 2In(a/ac)(a2J2T~)-1
.
(1.3.65)
Here a2 is a numerical factor of order unity,
a2 = r~d
f c{>(r)2dr.
(1.3.66)
Hence, the linear dependence of crc on a is replaced in this interval by the much slower logarithmic law.
For greater values of a, when In(a/ac) > (DT 0/r~)3/2, clusters with n:::= T~l cease to be exponentially
rare and our approximation is not valid.
The above results were obtained under the assumption that the breeding centers are long-lived, i.e. if
To ~ r~/ D. It can be shown that, in the opposite case of short-living centers with To ~ r~/ D, violation of
the mean-field approximation condition (1.3.46) is always related to the fact that the strong fluctuations
of the "space cluster" type with n :::= T~l cease to be exponentially rare.
To conclude this section, let us briefly discuss[18, 25] the fluctuational lowering of the explosion
threshold in a situation where fluctuations in breeding and decay rates are Gaussian,i.e. if
Ii = - Tn + DJn + g(r, t)n ,
(1.3.67)
where the random Gaussianfield g(r, t) has correlation functions
( g(r, t)) = 0 , ( g(r l' t1)g( r 2' t2)) = S exp(- r ~ 11r 1- r 21- T~ 11t1 - t21).
(1.3.68)
= 0) the explosion threshold in the model (1.3.67) is reached
at T = O. Fluctuations lead to the lowering of the explosion threshold.
The explosion threshold is sensitive to the relationship between the lengths 1= (DT 0)1/2and r o'
When r 0~ 1, it is given by
In the absence of fluctuations
(i.e. for S
T= STo,
(1.3.69)
.1"
irrespective of the dimensionality d of the medium.*) When 1~ r 0' we find
oj Note that in this case we can approximate
(1.3.69) coincides with the expression r
= Uo
the short-correlated
Gaussian field g(r, t) by a white noise with the overall intensity
for the explosion threshold found in section
Uo
= ST o' Then
1.2.
;
~"~..~~".
~
~
--~
==-
A.S. Mikhailov,Selected
topicsin fluctuational
kineticsof reactions
I
r=(Sr~/D)(I/ro)'
d=1;
r=(Sr~/D)ln(l/ro)'
d=2;
r=(Sr~/D),
335
d=3.
(1.3.70)
'Expressions(1.3.70) are valid at sufficiently small noise intensities S; the threshold value of r, given by
(1.3.70), should satisfy the condition rT 0 ~ 1. In the opposite case the explosion threshold is
determined by very rare strong positive bursts of the fluctuating field g(r, t). Note that, for the
stationary random field (i.e. when To~ 00), the explosion threshold is formally exceededat any value of
f. Indeed, in the infinitely extended medium, it is always possible to find such a strong positive
fluctuation of this stationary random field that it will lead to a local breeding rate that exceedsany given
value of r. Clearly, this result is an idealization. All realistic systemshave finite dimensions and,
therefore, there is a strongest fluctuation of the breeding rate inside them. Furthermore, the realistic
random fields are, as a rule, only approximately Gaussian, and the deviations from the Gaussian
properties become large for very strong fluctuations.
2. Population settling-down transitions in fluctuating media
2.1. Logistic growth in fluctuating media
Supposethat the population of certain particles (neutrons, free radicals, bacteria, etc.) is breeding in
a fluctuating medium. As long as the population density of these particles remains sufficiently small, its
growth follows a simple exponential law, which is typical of the explosive processes.At larger
population densities, the nonlinear effects should come into play. Usually they result in saturation of
the population growth. The simplest mathematical model, which describessuch saturation, is given by
the Verhulst (or logistic) equation
Ii = -an +fn - f3n2+ D.1n.
(2.1.1)
Here a is the death rate, f is the breeding rate, f3 is the nonlinear damping coefficient, and D is the
diffusion constant (or mobility) of the breeding particles.
When the breeding rate exceeds the death rate, the system obeying eq. (2.1.1) undergoes a
population settling-down transition. Below the transition point (for f < a), any initial population
becomesextinct as time elapses, while above this point even very small initial populations spread
eventuallyover the entire medium and give rise to a steady state with a certain population density, the
sameeverywhere in the medium:
n=O,
forf<a,
n=(f-a)/f3,
forf>a.
(2.1.2)
Sucha transition bears some similarities with the second-orderphasetransitions in condensedmatter
[26]. We can note, for instance, that the characteristic relaxation time, required to reach the steady
state (2.1.2), diverges as
trel ~ (fMoreover,
~,,~~
=
.f"
a)-I.
we can introduce
(2.1.3)
the correlation
length r c' defining it as the characteristic
~--_c:c
--
c_-
length scale of the