Article
Reference
Refined simulations of the reaction front for diffusion-limited two-species annihilation in one dimension
CORNELL, Stephen John
Abstract
Extensive simulations are performed of the diffusion-limited reaction A+B→0 in one dimension, with initially separated reagents. The reaction rate profile and the probability distributions of the separation and midpoint of the nearest-neighbor pair of A and B particles are all shown to exhibit dynamic scaling independently of the presence of fluctuations in the initial state and of an exclusion principle in the model. The data are consistent with all length scales behaving as t1/4 as t→∞. Evidence of multiscaling, found by other authors, is discussed in light of these findings.
CORNELL, Stephen John. Refined simulations of the reaction front for diffusion-limited two-species annihilation in one dimension. Physical Review. E , 1995, vol. 51, no. 5, p.
4055-4064
DOI : 10.1103/PhysRevE.51.4055
Available at:
http://archive-ouverte.unige.ch/unige:91476
Disclaimer: layout of this document may differ from the published version.
1 / 1
PHYSICAL REVIEW
E
VOLUME 51, NUMBER 5 MAY 1995Refined simulations of the reaction front for difFusion-limited two-species annihilation in one dimension
Stephen
3.
Cornell*Departement de Physique Theorique, Universite de Geneve, 2g quai Ernest Ans-ermet, CH 2211-Geneve g, Sisitzerland
(Received 17 October 1994)
Extensive simulations areperformed ofthe diffusion-limited reaction
A+B
—+0 in one dimension, with initially separated reagents. The reaction rate profile and the probability distributions ofthe separation and midpoint of the nearest-neighbor pair ofA andB
particles are all shown to exhibit dynamic scaling independently of the presence ofQuctuations in the initial state and ofan exclusion principle in the model. The data are consistent with all length scales behaving as t as t~
oo.Evidence ofmultiscaling, found by other authors, is discussed in light ofthese findings.
PACS number(s): 05.40.
+j,
02.50.—
r, 82.20.—
wI. INTRODUCTION
(4)Ba 02a
g2~
Ob
Ob
Bt 82X
where
R
is the reaction rate per unit volume and the diffusion constantD
has been assumed equal for both species. For the Boltzmann equation ansatzR =
kab, the solutionto
the resulting partial differential equations with the initial conditiona( — x,
o)= aoe(*) =
b(x,o),
(2)where 0 is the Heaviside function, has the scaling prop- erty
for
x((t
(3)with n
=
s[1].
This result may be understood by consid- ering the steady-state solutionsto Eqs. (1)
for boundary conditionsa( — x) -+
J~x~/D,a(x) -+
0, b(x)
i 0,—
and b(x)~ Jx/D
as x + oo. Under these conditions, there are opposing constant currentsJ
ofeither species and it can be shown [9,15] that the resulting reaction profile isof
the form'Present address: Department ofMathematics and Statis- tics, University of Guelph, Guelph, Ontario, Canada N1G 2VT1.
There has been
a
lotof
recent interest inthe scaling be- haviorof
the reaction front that exists between regions of initially separated reagents A andB
that perform Brow- nian motion and annihilate upon contact according to the reaction scheme A+ B -+
0 [1—18].
The evolutionof
the particle densitiesa(x, t)
and b(x,t)
at positionx
and timet
is governed by the equationswith A
= 3.
Returning to the the time-dependent case, the quantity(a —
b) obeys a diffusion equation, whosesolution for initial conditions (2)is
~/(2/Dt)
b
— a=
exp( — y')
dy.7r Q (5)
I
et us assume that the reaction takes place within a regionof
width mt,
with o.( 2.
The profilesfor ie
« x « t
~ areof
the form a oc aox/t~,
sothere is a current
of
particles arrivingat
the origin of the formJ = DB
a~ t
~.
The characteristic time scale on which this current varies is (dinJ/dt) i
oct,
whereas the equilibration time for the &ont is of order (m/D) t « t.
The front is therefore formed qua- sistatically and soEq.
(3)may be obtained from (4)sim- ply by writingJ
cxt
Simulations and experiments appear
to
confirm these results when the spatial dimension d is two or greater [2—4,6].
In dimension less than two, strong correlations between the motionsof
the two species cause the Boltz- mann approximationR =
kabto
break down. However, the solutionto
the steady-state problem is still of the form(4),
albeit with a difFerent exponent A= 1/(d + 1)
[15,
17]. If
the results from the steady-state may still be used, this would lead againto
dynamical scalingof
the form(3),
with cz=
4 in d= 1.
Simulations using a one-dimensional probabilistic cellular automata(PCA)
model appearedto
verify the dynamical scaling form(3),
though with n
= 0.
293+ 0.
005[5].
Monte Carlo simula- tions also found o.0.
30+ 0. 01 [8].
However,
a
recent article by Araujo et al [16]haschal-.
lenged the validity of the scaling form
(3).
This article reported Monte Carlo (MC) simulations in one dimen- sion, using an algorithm where the A andB
particles always react on contact and so are unable to cross over each other. The rightmost A particle (RMA) is there- fore always to the leftof
the leftmostB
particle(LMB).
Defining
l~~
as the separation between the RMA and1063-651X/95/51(5)/4055(10)/$06. 00 51
1995
The American Physical SocietySTEPHEN
J.
CORNELI.the LMB and m as the midpoint between them, Araujo et a/. found that the probability distributions P~ and
P
ofthese quantities displayed dynamic scaling, with char- acteristic length scales
t
~ andt ~,
respectively. Mean- while, the different moments of the reaction profile were described by a continuous spectrum oflength scales be- tweent
~ andt
~.
More speciIIically, defininglfq l~Ii P((l~gg)dlgii
~1/9
/m[
P
(m)dm / (7)R(x, t)dx
)
Araujo et al. found that t~q~
t
~ and m~q~t /,
butx~q~
t
& with 4 & o.q & 8 increasing monotonicallywith q. They also proposed the following form for
B:
which predicted values ofo.qthat were in good agreement with their numerical findings (the prefactor
t
i~4, essen- tial for consistency, is missing in[16]).
The authors of[16]argued that Poisson noise in the initial state causes the reaction center
to
wander anomalously as m oct
/ invalidating the use of the steady-state results.In this paper,
I
first describe extensive simulations of this system, using two independent models the PCA model used in [5] anda
MC model similarto
that of Araujo et al. in[16]. I
find that dynamical scaling ap- pearsto
hold for P~,P,
andB
independently of the existence of Poisson Huctuations in the initial state andof
the presence of an exclusion principle. WhileI
con-firm the result 4t~q~
t ~, I
find instead that both m~q~and x~q~ appear to scale as
t,
with o.0.
28+ 0.01,
independently of q. The high statistics and wide time domain accessible in the PCA simulations show that this exponent is decreasing monotonically in time, consistent with the asymptotic result o.
=
4 predicted by the anal- ogy with the steady-state result. The measured forms of P~,P,
andB
are foundto
be described by very simple analytic forms to high accuracy.I
then discuss the va- lidity of the fluctuation argument used by Araujo et al.to
explain the result mt
/.
An exact calculation of a related quantity suggests that the wandering of the reac- tion center should instead beof
the ordert ~,
which isnot sufBcient
to
make the useof
the steady-state analogy invalid. Someof
these results have been discussed in a previous paper[18].
j:I. MONTE CAB.
LDMODEL A. Description of
xraadelThe model described in Ref. [16]consists of indepen- dent random walkers with no exclusion principle. In the
interests of computational eKciency,
I
used amodel that is identical, provided the site occupation number is nottoo
large, but whose site updates may be effected using alookup-table algorithm. In this way, it was possible to obtain statistics equivalent to the simulations in [16]in the space of a few days.The model has an "exclusion principle", in that no more than 2l& particles ofeach type are allowed per site.
In the diffusion step, each ofthese particles moves onto
a
neighboring site in sucha
way that no more thanI„
particles may move from a given site in the same direc- tion at once. This constraint automatically satisfies the
"exclusion principle.
" If
there arel„or
fewer particles ona site, then the direction in which each particle moves is chosen independently
at
random.If
there are more than L„particles, the same redistribution method is used for the "holes"i.e.
, the probability ofj
particles moving tothe right when the occupation number is k is the same as
(/„— j)
particles movingto
the right when the occupa- tion number is(2l„— k).
The diffusion constant for this model is —,'.
In these simulations, the value
I„=
13was used (thiswas the largest value that could be implemented eK- ciently). Since the average density was 1 or less, the frequency ofevents where the occupation is greater than
t„
is oforder e/(l„+ 1)!
4x 10,
so these events are extremely rare (the simulations represent 21000sam- ples of 4000sites over 25000time steps, so the expected total numberof
such events is less than10).
The influ- ence ofsuch events on the results isstill smaller since the probability of a large number of particles spontaneously moving in the same direction is low (e.g., 14 indepen- dent walkers move in the same direction with probabil- ity 2 is=
104).
Moreover, the universality class for the scaling properties is not expectedto
depend on such events, as the reactions take place in the zone where the density is low. These results may therefore justifiably described as equivalentto
those reported in[16].
TheFQRTRAN implementation of this algorithm performed
1.
4x 10 site updates per second on a Hewlett-Packard (HP) 9000/715/75 workstation.One time step consists ofmoving all the particles, fol- lowed by a reaction step. The pure diffusion algorithm has a spurious invariance in that particles initially on even sites will always be on even sites after an even num- ber oftime steps and on odd sites after an odd number oftiine steps (and contrarily for particles initially on odd sites). In accordance with the prescription in [16]that an A particle never be found
to
the right of aB
particle, it is important that the reaction takes account of parti- clesof
difFerent types crossing over each other(i.e.
, an A particleat
sitei hopping toi+
1at
the same time that a H particle ati+1
hopsto
sitei).
The reaction algorithm first removes such particles and then annihilates any re- maining AandB
particles occupying the same site. This isillustrated inFig. 1.
Four sites are shown, with initially two sites occupied by A particles (represented above the line, labeled 1—5) and two occupied byR
particles (be- neath the line, labeled 6—10).
In the difFusion step, each particles moves ontoa
neighborat
random, producing the state(ii).
The reaction first deletes the A— R
pairREFINED SIMULATIONS OF THEREACTION FRONT
FOR. . .
0~
01 03
0~
03 04
6 8 i 6 8
Q9
I
Qo
'
, 0'
Q7
05 ', 03
00
0'(=1000 1.0
~-0.8 0.6 0.4
I
Diffusion
(I) (ii) Delete
crossing
Delete
(iii) (IV)
cohabiting
0.2 0.0
3.5 4.0
logi&(t)
2 1/2 3 4
FIG. 1.
Illustration ofthe Monte Carlo reaction-diffusion algorithm, showing the diffusion step and the two stages of the reaction step that erst remove particles that have crossed over and then react those that are at the same site.FIG.
2. Scaling plot ofthe density difference a(x, t)—
b(x,t)for the MC data with random initial condition. Inset: bias plot for the total number ofCparticles
Zc = j C(x,
t)dx.(4,6) that crossed over, leading to state (iii), and then removes the pairs that sit at the same site, giving afinal state
(iv). If
the "delete crossing" step were not present, the reaction step would simply remove the pair (4,8)fromstate (ii),
leading to a state where there areB
particlesto
the left ofAparticles.B.
Simulation resultsthe values they would have for an infinite system. Figure
2shows aplot
of
(a—
b) as a functionof
(x/ti~ ) for threetime values, displaying excellent rescaling. Asimpler test
of
finite-size effects isto
show that thetotal C
particle numberKc = j C
dx is proportional tot
~ . The insetto Fig.
2 confirms thatZ~t
~ is indeed independentof
time.Figure 3is a log-log plot of
X,
,m,
, andl,
as a function of time, whereX«) =
C(x, t)
dx)
(10)
where
C— : j Bdt
is the profile of the reaction product.This quantity differs from
x(q),
but sincej Cdx
oc ti~2and provided x('il behaves as a power of
t, Eq.
(5) of [16]shows that they should have the same scaling behavior.
From
Eq. (1),
the diff'erence in the particle densities(a —
b) obeys a simple diKusion equation, whose solution is given by(5).
Any finite-size effects in the data would first show up in deviationsof
the particle profiles from An approximation to a Poissonian initial stateof
av- erage density unity was prepared by performing 16 at- tempts to add an A particle, with probability1/16,
to eachof
the first 2000 sites of a 4000-sitelattice.
The other half of the lattice was similarly populated withB
particles. At the boundaries, particles that attempted to leave the system were allowed
to
do so, but a random number (distributed binomially between 0 and 16,aver- age —) of particles was allowedto
reenter the system at the end sites. The average density at the extremities was thus keptat
the value unity.In order
to
mimic the simulations in [16]as closely as possible, instantaneous measurements were madeof l~~,
m, and the concentration profiles of the product and reagentsat
times 1000, 2500, 5000, 7500,.. .
,25000.
These were then averaged over 21000independent initial conditions. The quantities II'(q) and m(q) were measured from the probability distributions over the samples and a quantity X(q) was de6.ned as
(11) (12) (13)
1.5 Q
1.3
1.1
CD
~0.
bQ 93.0 3.5 4.0
log
10(t)~1,(1) (2) l(8) X
l(16) fjrn+(2)
(8)
&&In~
~m,(16)
x"'
Ox,(8) (16)
FIG. 3.
Iog-log plot ofI,
,m, , andX,
(see the text)from the MC simulations with Poisson initial conditions.
and
(~,
p~, and A~ are constantsthat
will be defined later. The straight lines are fitsto
the last eight points forX,
(2),m,
(2), andl,
(i). The gradients for least-squares Gtsto
the curves inFig.
3 are listed in TableI.
The exponent describing I(~l isclose to —,as was found in[16].
However, the results for m( )and
X(
) differ dramatically from those ofAraujo et a/.First,
the exponent describingm(t)
appears to be closeto 0.
29,insteadof 0.
375as they found. Second, the exponents describing X(q) appearto
be independent of q. This means thatC(x, t),
and4058 STEPHEN
J.
CORNELLTABLE
I.
Comparison of the simulation results in this paper for the Monte Carlo model (MC) and the PCA model with Poisson (PCAP) and full (PCAF) initial conditions with those ofAraujo et al. (ALHS) [16].Numbers in parentheses represent the statistical error in the preceding digit.Result size
exclusion principle?
initial density initial state averaging max time exponents I(~)
~(j.6)
m(')
x(')
~(8)
(&6)
fit over last..
.
ALHS 2000 no
1.
0 (uiiiform) 6000—15000 25000 0.25 0.25 0.375 0.375 0.312 0.359 0.367theoretical value
MC 4000 no
1.
0 Poisson 21000 25000 0.251(3)O.23(1) 0.281(4) 0.29(1) 0.2799(2)
O.282(2)' 0.30(2) 8points
PCAP 4000 yes
0.5 Poisson 82176 102400 0.2510(6) 0.248(3) 0.287(1) 0.284(l) 0.286(2) 0.280(4) 0.28(1)
5points
PCAF infinite yes 2.0 uniform 64000 409600 0.2542(4) 0.2609(3) 0.300(1) 0.299(3) 0.291(1)
O.293(1) 0.293(2) 6points Seethe text.
Measured from
X
by implication
B(x, t),
obeys a simple scaling form, in contrast to the anomalous form(9).
To investigate for a trend in the exponents describing these quantities, the efFective exponent (defined as the gradient between successive points in
Fig.
3) is plotted as afunction of 1/ logio(t) inFig. 4.
The data for I( )and m~ ) are far too noisy for any information to be obtained.The exponent for X~ ~ appears
to to
decrease slowly in time, but the time window in these simulations istoo
narrow for conclusive deductions to be made.Figures 5, 6,and 7 are plots of
C, Pt(l~~),
andP (m),
respectively, as a function of appropriate scaling vari- ables,
to
show the subjective qualityof
scaling for these quantities. The profilesof P
(m) and PI(l~t3) suggest the following forms:PI(l)
= —
exp/Ib'
m«~
= &-'m„ (16)
where
~I
+
(yq+i(i/m/2) /q for q odd [(q/2)!] for q even.
(17) (18)
These forms predict the following results for the moments ofthese distributions:1
/'m)'
P
(m)=
expmp K mp (14)
Using these values
of
pq and Aq inEqs. (11)
—(13),
onewould expect
m,
andI, to
be independent ofq if the forms(14)
and(15)
are valid. The coincidence of the curves inFig. 3
confirms this.0.40
g
0.35C
~
0.30~M
~~0.25 om(2)
~X(2) 01(l)
0.6
H 0.4
0.2
~t= 1000 o t=~000 +t=25000
0.20 0.00
1/log I
0(t)
0.35 0.0-4 o(2) 2
FIG.
4. EfFective exponents forI, m,
andX
(seethetext) from the MC simulations.
FIG.
5. Scaling plot forC(x, t),
forthe MC simulations.REFINED SIMULATIONS OF THE REACTION FRONT
FOR. . .
4059 0.80.0
~MC,t=1000 fjMC,t=5000 x MC,t=25000
~ PCAP,t=1600 o PCAP,t=12800 gPCAP,t=102400
~PCAF,t=800 a PCAF,t=6400 x PCAF,t=51200
%PCAF,t=409600
0
-2C46
0
-4C4
I
aa-6
0
MC
'0jrp
. . .
(
i~0,g
:~. R
andP
-&nglitV
1 P1,MC
1 P1,PCAP
1 P1,PCAF
P,
MCP,
PCAPP,
PCAFR, PCAP R,PCAF RM&
FIG.
6. Scaling plot for Pt(I), for the MC simulations (MC) and the PCA simulations with Poisson (PCAP) andfull (PCAF) initial conditions.
Figure 8 is an explicit test of the forms
(14)
and(15)
against the data, by plotting logro[m()P
(m)] and logio[l( )lPi(l)]
against (m/m())
and (l/l()),
re-spectively, at
t =
25000.
TheY
ordinate has been shifted sothat all curves are coincidentat
the origin. The curve labeled RM~ islogio[C(x,
25000)— C(x,
22500)],
which is approximately proportional
to R(x,
25000),
asa
functionof (x/I( ))
. The straight line for this curve suggests that the reaction profileR(x, t)
is alsoa
Gaus- sian. This again contradicts the form(9)
proposed by Araujo et al.It
is not possible, however,to
derive ana- lytical forms for (~ that leadto A,
being independentof
q without assuming a form for x('i)(t) for allt,
so the values of (~ used inEq. (11)
were chosen numerically in an ad hoc fashion.III. PROBABILISTIC CELLULAR
AUTOMATAMODEL
A. Description of
model0.5
o~[Qx
)
~MC,t=1000 oMC,t=5000 x MC,t=25000
~PCAP,t=1600 o PCAP,t=l2800
~ PCAP,t=102400
This model has been described extensively in previous publications [5,
19].
In the one-dimensional realization of this model, there are upto
two particles ofeach species at each site, labeled by the direction from which they moved10 20 30
2 2 2
l
orm orx
FIG.
8. Fits ofP,
Pt, andR
to Eqs. (14),(15),and (25)from the MC simulations (MC) and PCA simulations with Poisson (PCAP) and full (PCAF) initial conditions. The A axis is rescaled and the Yaxis isshifted for clarity. For curve RM~, see the text.
onto the site
at
the previous time step. The diffusion step consistsof
changing the velocitiesof
these particles and then moving the particles onto the neighboring sites ac- cordingto
their new velocities.If
there are two particles per site, they both move in opposite directions, whereasa
single particle will change direction with probabilityp.
The value used in these simulations was p
=
2, sothat
the particle forgets its previous velocityat
each time step and the model is equivalentto
the MCmodel with lz— — 1.
The reaction step consists ofchecking each site for si- multaneous occupancy
of
A andB
particlesat
the startof
the time step and removing any pairs that hopped onto the site from opposite directions. Using the segre- gated initial condition and this "infinite" reaction rate,a
site can only be occupied by anA-B
pairif
the A ar- rived from the left and theB
from the right. This model has the same two-sublattice structure asthe Monte Carlo model defined above and this is preserved by the reac- tion algorithm, so these two sublattices must be viewed as two independent systems. There is therefore an inde- pendent nearest-neighbor A-B'pair for each sublattice. A multispin-coding implementationof
the algorithm simu- lates 64 independent systemsat
once.The quantities
P (m),
P&(l), andR(x, t) at
measure- ment timet
were estimated by averaging over the intervalt(1 —
b)( t ( t(1 +
b), with b= 0. 05.
We may estimatethe order
of
magnitudeof
the systematic error that this introduces into the measured shapeof
these quantities.Let
F (x, t)
be the estimateof
afunction I"(x, t)
using the above method. Then0.0
~PCAF,t=800
~PCAF,t=6400 x PCAF,t=51200
)KPCAF,t=409600
I
FIG.
7. Scaling plot forP
(m), for the MC simulations (MC) and the PCA simulations with Poisson (PCAP) and full (PCAF) initial conditions.1 ~(~+~)
P— : t(x-b) E'(x, t')
dt'
t(1+8)
I
&(x, t)+ (t'- t) — +(x, t)
2tb, (,
~)(
' Ot1,
2 c)2+ (t' — t)' F(x, t) — + .
~dt'
2 c)t2
(tb)2 c)2
= +(x, t) +» E'(x, t) + O((tb)').
(19)
(20)
(21)
4060 STEPHEN
J.
CORNELL The fractional error is thereforeof
order (tb)E/(6E).
This systematic error has no effect on the scaling be- havior however
If
.E(x, t) =
tsar(x/t),
we have(22) (23)
0.6
4=1600 0.5
0 4 0.3
~
0.2 where P(y)=
(2h)f
& 0 P(yo )de, soE
has thesame scaling properties as
F.
Inorder
to
maximize the statistics, the reaction profile A was measuredat
every time step betweent(l —
h) andt(l +
h). However, the quantities m and lare much more cumbersome to measure using this prograin (due to the multispin coding) and so were only measured every ten time steps. No significant loss in statistics is incurred since these quantities have very strong time autocorre- lations. The FORTRAN implementationof
this algorithm performed3.
7 x 10 site updates per second on a HP 9000/715/75 workstation.0.1 0.0
3 4 5
log 10(t) 1/2 3
hypothesis and the forms for the scaling functions
(14)
and(15)
and also that the reaction rate profile has a Gaussian formFIG. 9.
Scaling plot ofthe density difference a(x, t)—
b(x,t)
for the PCA data with random initial condition. Inset: bias plot for the total reaction rate Ztt
= f Bdx.
B.
Simulation resultsR(T, t) =
exp—
( —
)
(25)1. Poi
saon initial conditionx+
() —
pox (24)and p~ is the appropriate scaling factor for Gaussian dis- tributions [see
Eqs.
(12) and(17)].
The curves for m,„
have been shifted vertically (by
0.
2) for clarity; other- wise they would betoo
close to the curves forx,
. The straight lines are afitto
the last five points, forthe lowest values ofq. The gradients ofleast-square fits for all the curves are summarized in TableI.
The collapse of the curves for different values of q confirms both the scaling An initial condition with Poisson-like density Auctua- tions was prepared by filling eachof
the appropriate site variables (A particles forx (
0,B
particles forx )
0)with probability
4.
The lattice size was 4000 sites and at the boundaries particles were freeto
leave the system, with the density at the boundary maintainedat
an aver- age value of 2 by allowing A particles to enter from the left andB
particlesto
enter from the right, randomly with probability —,.
Measurements were takenat
times 200—102 400time steps, with the interval between mea- surements doubling progressively. The quantitiesP (m), P
(l)andB(x, t)
were measured as described above and then averaged over 82 176 independent realizations of the system. The quantities m~~~, l~~~, and x~~~ were then measured from the (1/q)th powerof
the normalized qth moment ofthese quantities.Figure 9shows a plot
of (a —
6) asa functionof
(x/ti~2)for three time values and
a
plotof Z~t
~ (whereE~ =
f Bdx)
asa
function oft.
These plots showthat, just
asin the MC simulations, there are no finite-size effects.
Figure 10is a log-log plot
of x,
,m~,
andl,
as a function of time, where~2.
01.5
CO
1.0
~
bQO05
1 (q)
I m,
(q)(p
I I I
2 3
l
4 5
1Og10(t)
gl(1) )(2) e&,
)(g) X
)(16)
nm,(2)
om,(8)
~m,(16)
~X4(2) (11) OXt, +Xt,(16)
FIG.
10. Log-log plot ofl,
, m,~}, andx,
(see the text) from the PCA simulations with Poisson initial conditions.The curves for m have been shifted vertically for clarity.
Figure
11
shows the effective exponents for x~ ~, m~2~, and l~ ~, from the successive gradients inFig. 10.
The curves are much less noisy than those inFig.
4,by virtueof
higher statistics and the use of coarse-grained time averages. There isa clear trend for the effective exponent for x~ ~to
decrease as time increases, consistent with the asymptotic value 4 predicted elsewhere [15,17].
The exponent for m~ ~ appearsto
increase initially, but the last few points appear alsoto
decrease and in any case an asymptotic value0.
375is ruled out.The rescaled forms
of Pt(l), P
(m), andB
are de- noted byP
CAP in Figs. 6, 7, and 12 respectively.Figure 8 shows
a
fit ofPt(l), P
(m), andR
to the forms(14), (15),
and(25).
PromEq. (19),
usingE(x, t) =
At ~exp(—
Ax /t),
the fractional error intro- duced by the coarse-grained time averaging isfound to be(b
/6)(x/ur),
where tU= (f x Edx/ f Edx).
The51 REFINED SIMULATIONS OF THE REACTION FRONT
FOR. . .
4061 0.32~
0.300
~
0.28~W
~ 0
0.26om(2)
~X(2)
0)(I)
C) p
Ei
0.24
0.0 0.2 0.4
I/log 10(t)
FIG. 11.
Effective exponents forI, m,
and x (seethe text) from the PCA simulations with Poisson initial con- ditions.
g1„(I) )(2) )(8) X
)(l6) om,(2)
om,(8)
~m,(16)
~X~(2) C X~(8) +X~(16) C' 1.4
—
l (q)
(q) a q
'
m,
8 (q)q X~
I
3
0
~
0.6~ 0
0.22 4 5 6
log
10(t)FIG.
13.Log-log plot ofI,
,m, , andx,
(see the text) from the PCA simulations with full initial conditions. The curves for m have been shifted vertically for clarity.measurement of these quantities is therefore expected to be accurate for the first four decades or so as is indeed observed.
2.
Eull initial condition8Because
of
the exclusion principle in the PCA model, the system is completely static in regions where the oc- cupation number is zero for one species and assumes its maximal value for the other.If
one starts from a lattice that is filled with A particles upto x =
0and filled withB
particles forx )
0,simulations may be speeded up by only updating the lattice inthe region where a "hole" has penetrated.By
checking explicitly that such holes never reach the physical boundary of the system, it is possi- bleto
perform simulations on a system that is effectively infinite, thus having no finite-size effects.Simulations
of
64 000independent evolutions ofa
full lattice were run for 409 600time steps. Measurementsof
P (m),
P~(l), andB
were made using the same method as for the Poisson initial condition. Results for these simulations are shown in Figs.13
and14. It
might be expected (considering the arguments in[16])
that thiscase would be in a different universality class &om the case with randomness in the initial
state.
However, the results for the exponents (see TableI)
are very close to those measured for the caseof
Poisson initial conditions and the marked decrease of the exponents for x(2) (ar- guably towards0.
25) is also seen inFig. 14. It
is inter- estingto
note that the transient trends in m~ ~ and l~ ~ are in the opposite senseto
the Poisson case.Scaling plots for P~,
P,
andB
are shown in Figs. 6,7,and 12,denoted by
PCAF.
Plots of/ P~,P,
andB
may be found in
Fig.
8,confirmingthat
the profiles again have the forms(14), (15),
and(25).
Figure 15is
a
plot ofm~ ~ and x~ ~ asa
functionof
the time-dependent currentZ~ = f Rdx,
from the simula-tions both with
(P)
and without(F)
Poisson fluctuations in the initialstate.
The two curves for x~ & are almost coincident, which is what would be expected if the reac- tion profile depended upon the current only. The curves for m~ ~,however, are not quite coincident, showing that this quantity is more sensitiveto
the initial condition.Incidentally, numerical tests showed that the diffusion current
at
the origin has Poissonian noise whether the initial state contained such Huctuations or not.0.5
~F, t=800 00 1200 09600 600 2800 02400
0.33
~
0.310
(g0.29
~W
~ 0
~0.
27 om(2)~X(2)
O)(I) .~
o.&
0
4
0.0 0.25
0.0 I/log 10(t) 0.4
FIG.
12. Scaling plot for R(x,t),
for the PCA simulations with Poisson (P)and full(F)
initial conditions.FIG.
14. Effective exponents forI, m,
and x (seethetext) from the PCA simulations with full initial conditions.
4062 STEPHEN
J.
CORNELL 1.80.2
op,x(2)
Consider the gradient of p:
a. p=) 2(~t)-'&'
t )
x+
y,f (x+ y)'l )
(32) s —
i
2=
2—
p(x,t) + )
(7rt ) '~x;exp ~— )
+ +y,
expI— (x+
y;)-3 -2
"gio(~R)
FIG.
15. Plot of x and m against the time-dependent current for the two sets ofPCA simulations.IV. THE EFFECT OF POISSONIAN FLUCTUATIONS
INTHE INITIAL STATE
The measured value mt
~ in Ref. [16]was justified by an argument about the Poisson Huctuations in the initial condition. The argument went as follows: after timet,
particles within a distancet
~ have had a chance of participating in the reaction. The number of particles within a distancet
/ is of ordert
~+ ct
/ Since each reaction event "kills" precisely one A and oneB,
there is therefore a local surplust
~of
one of the species. The majority species therefore invades the minority species by a distance m, such that the numberof
minority particles between the origin and m is Since the particle profiles vary likex/t ~,
this means thatInj (x/t
~ )dxt
~ )so m~ t
~order
to
assess the validity of this argument,it
is possibleto
apply it toa
related quantity upon which ana- lytical calculations may be made. Consider the diffusion equation Bqp(x,t) = 48
p(x,t)
in one spatial dimension, with an initial condition that consists ofarandom seriesof
negative Dirac bpeaks forx
&0 and positive Dirac b peaks for x) 0. That
is,f
p(z o)( z'i ( 2zxpi
Suppose that xo
t,
wherea
isexpectedto
be less thati,
and let e=
ts+li~2l, with 0(
b(
(——a).
Then thecontribution to the integral in
(33)
for ~x~)
e is oforderexp(
—
e /t) exp(— t ),
which vanishes ast
+oo. How- ever, for ~x~(
e, the argumentof
the second exponential has an upper bound 2exp/t + 0 and so the asymptotic value of the integral isfound by using the first few terms only of the Taylor expansion ofthis exponential. In other words, the leading contribution to xo as t~
oo is givenby
OD 2
p(z, o)e ~
dz—
t
p(z, 0)ze ~ dz= 0.
(34)X~
~
~
~
1~i2
~ ~
~The expectation value of the second moment ofxo is
x2 2
t2
j
p(x, o)e ~ dxj
p(y, o)e "~ dyxp
(f xpjx,
OIe'
dTgyp(y,
O)e"
dy (35)The second term on the right-hand side of
Eq.
(32) is strictly positive. The gradient of p when p is zero is therefore strictly positive, so, since p iscontinuous for allt )
0, p is zeroat
precisely one point, sayxp(t).
It
is possibleto
find the probability distribution of these zerosP(xp)
over the ensembleof
initial states. The position xp is defined by p(xp,t) =
0or, equivalently,p(x
t=O) =)
i=1b(x-x) — )
i=18(x+y),
((*, o))= ()
(p(x,
0)p(y,0)) =
sgn(xy)+
h(x—
y), (27)(28)where
x; )
0,y,) 0. If
the intervals between thex,
andy; have a Poisson distribution, one has
To evaluate this average, write
p(x,
o)=
sgn(x)+
w(x), where (w(x))=
0 and(r(x)r(y)) =
b(x— y).
Thenj xp(x,
0) exp( x/t)
dx— = t+ j x~(x)
exp(—
x2/t) dx, thesecond term being typically much smaller than the G.
rst.
To 6.nd the leading contribution to xo, it is sufficient
to
replacej xp(x,
0) exp(—
x /t)dx in the denominator byt.
We therefore have
p(x,
t) = ( x — x' ')
p(x',
O)(~t) -'~'
exp~
—
~ dx' (29)
t
~x
— x
~'= )
(art) i~2 exp ~— '
where ( ) represents an average over the variables
x,
,y,.
The solution for p may be written in the formCK) CX)
(*p)
=
4
(p(x
o) p(yo))
x'+ y'5
xexp
~——
~dxdy+.
e 2-'t
dx+ --.
4
(36)
(*+ y')'&
t
(3o)7rt
+
0~~2 8
(39)
51 REFINED SIMULATIONS OF THE REACTION FRONT
FOR. . .
4063 Similarly, the 2nth moment ofP(xp)
isof
the form2n
(xp") =
— dxg. dx2„exp ~—
x(p(zi,
0) 'p(z2„, 0)) + . .
(40) (2)!
& t&"~
2-.
~~ 8+
0~~(41)
For a distributionof
the formP(xp)
g(A/vr) exp(—
Ax2p), one has(42) A comparison with (41)gives
P(zp) =—
2 exp— —
8 xo 7rt)
1.25 A
CO
V+0'75
bQO
~~0.25
-0.25 0
X
X x x
b, b.
0 0
0 0
0 0 o 0
0 0 0
logl0(t)
FIG.
16. Log-log plot of(x~p) versus t for q=
2 (o),4 (o), 8(Z),
and 16(x),
from numerical solutions of Eq. (30).The straight lines are the asymptotic solutions from Eq.(41).
The distribution ofxo istherefore characterized by a sin- gle length scale A /' (xg /'
.
Figure 16 shows the moments of Po, averaged over 10 000realizations, from anumerical solution of the zero
of
p fromEq. (30),
compared with the asymptotic pre- dictions ofEq. (41).
From
Eq. (1),
the density difference(a —
b) in thereaction-difFusion problem, averaged over evolutions, obeys asimple difFusion equation. The quantity p with the initial condition (26) is therefore equal
to (a —
b) forthe initial condition with Poisson fluctuations used in the numerical simulations, with negative peaks correspond- ing
to
A particles and positive peaks correspondingto B
particles. The quantity xp differs from m(t) because the latter contains further fiuctuations due
to
the difFusive noise that has been averaged over in the former. How- ever, the argument used in [16] to obtain mt
~ may be applied equally wellto xo.
The reaction center shiftsto
compensate for a local majority ofordert
/ in oneof
the species and the argument predicts xot
~. It
is interestingto
note that the correct exponent is obtained if the initial valuea(x,
0)=
ap is used instead of the valuea(z, t)
ocz/t
~at
timet
in the balance equationf ' a(x, t) t
~ . This ambiguity is probably the reason for the argument being incorrect.V. CONCLUSIONS
It
appears from extensive simulations that the reac- tion profile in this system has the same simple dynamic scaling form independentlyof
the presenceof
an exclu- sion principle and ofrandomness in the initialstate.
The motionof
the reaction center dueto
the Poisson noise ap- pears onlyto
account fora contributionof
ordert
~to
the reaction width, which isnot large enoughto
alter the scaling behavior. The measured exponent= 0.
29 describ- ingboth the reaction width and the midpoint fluctuations appears in factto
be decreasing slowly in time, with fa- vorable evidence for an asymptotic value0.
25. This, to- gether with the measured form for the reaction profile, is consistent with the steady-state results being applicable [15,17,20].
It
is, nevertheless, surprising that the approach to the asymptotic behavior should be so slow.It
is not clear whether logarithmic corrections should be present, as they do not occur in the steady-state problem[17].
How- ever, in these simulations the ratio of the reaction widthtp to the diffusion length
(Dt)
~~2 was never smaller than— 0.
2, whereas the applicationof
the steady-state argu- ment requires that this ratio be small. This could ac- count for the fact that the asymptotic regime has not been reached. Simulations where this ratio is truly small would not appearto
be practicalat
present.An investigation
of
the simulation procedure used by Araujo et a/. has revealeda
few errors in the results pub- lished in[16].
A repeatof
their simulations appearsto
confirm the resultsof
the present article forP
and P~and the behavior m
t,
but does not findthat R
satisfies ascaling ansatz
[21].
This inconsistency between my results and those of Araujo et al. is currently unex- plained.A recent calculation by Rodriguez and Wio [22] sug- gests that the reaction profile
R
should. be the superposi- tion oftwo scaling forms, with width exponents 3 and 8, respectively. However, these exponents and the form they predict forR (
exp[—
(z/m)s~2]) do not agree with the resultsof
simulations. The approximation scheme they used would therefore not appearto
be valid, unless it describes aregime inaccessibleto
simulations.The simulation evidence in favor
of
d.ynamic scaling in this model is very strong. However, the numerical evidence that all length scales scale asymptotically ast
~ is far from conclusive and so needs to be put ona
sound theoretical basis either by an exact calculation or bya
rigorous justification for the analogy with the static case.ACKNOWLEDGMENTS
I
would liketo
thank Michel Droz, Hernan Larralde, John Cardy, andBen
Lee for many interesting discussions and Michel Droz for acareful readingof
this manuscript.I
would also liketo
thank Mariela Araujo for making detailsof
her simulations availableto
me.STEPHEN
J.
CORNELL [1][3]
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