Refined simulations of the reaction front for diffusion-limited two-species annihilation in one dimension

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 1995

Refined 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 and

B

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.

—

_w

I. INTRODUCTION

(4)

Ba 02a

g2~

Ob

Ob

Bt 82X

where

R

^is the reaction rate per unit volume and the diffusion constant

D

has been assumed equal for both species. For the Boltzmann equation ansatz

R =

kab, the solution

to

the resulting partial differential equations with the initial condition

a( — _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 solutions

to Eqs. (1)

^{for boundary} conditions

a( — _{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 currents

J

^ofeither species and it can be shown [9,15] that the ^resulting reaction profile is

of

the form

'Present address: Department ofMathematics and Statis- tics, University of Guelph, Guelph, Ontario, Canada N1G 2VT1.

There has been

a

lot

of

recent interest inthe scaling be- havior

of

the reaction front that exists between regions of initially separated reagents A and

B

that perform Brow- nian motion and annihilate upon contact according to the reaction scheme A

+ B -+

_{0 [1—}

18].

The evolution

of

the particle densities

a(x, t)

and b(x,

t)

at position

x

and time

t

is governed by the equations

with A

= _3.

Returning to the the time-dependent case, the quantity

(a —

_{b) obeys a diffusion equation, whose}

solution 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 region

of

width m

t,

^{with o.}

( ^2.

^{The profiles}

for ie

« ^x « ^t

^{~ are}

^of

^{the form a oc aox/t}

^~,

^so

there is a current

of

particles arriving

at

the origin of the form

J ^{= DB}

^a

^{~ t}

^~

^.

The characteristic time scale on which this current varies is (din

J/dt) i

_oc

t,

whereas the equilibration time for the &ont is of order (m

/D) t « ^t.

^{The front is therefore} formed qua- sistatically and so

Eq.

(3)^may be obtained from (4)^sim- ply by writing

J

^cx

t

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 motions

of

the two species cause the Boltz- mann approximation

R =

kab

to

break down. However, the ^solution

to

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 again

to

dynamical scaling

of

the form

(3),

^{with cz}

=

₄ in d

= 1.

Simulations using a one-dimensional probabilistic cellular automata

(PCA)

model appeared

to

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]^has

chal-.

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 and

B

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 left

of

the leftmost

B

particle

(LMB).

Defining

l~~

^as the separation between the RMA and

1063-651X/95/51(5)/4055(10)/$06. 00 ⁵¹

1995

The American Physical Society

STEPHEN

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

^~ and

t ~,

respectively. Mean- while, the different moments of the reaction profile were described by a continuous spectrum oflength scales be- tween

t

^~ and

t

^~

.

More speciIIically, defining

lfq 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 /,

but

x~q~

t

^& with ₄ & o._q & ₈ increasing monotonically

with 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 oc

t

^/ 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] and

a

MC model similar

to

that of Araujo et al. in

[16]. I

find that dynamical scaling ap- pears

to

hold for P~,

P,

^and

^B

independently of the existence of Poisson Huctuations in the initial state and

of

the presence of an exclusion principle. While

I

^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,

^and

^B

^{are found}

^to

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 m

t

^/

.

An exact calculation of a related quantity suggests that the wandering of the reac- tion center should instead be

of

the order

t ~,

^{which is}

not sufBcient

to

make the use

of

the steady-state analogy invalid. Some

of

these results have been discussed in a previous paper

[18].

j:I. MONTE CAB.

LD

MODEL A. Description of

^xraadel

The 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 not

too

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 such

a

way that no more than

I„

particles may move from a given site in the same direc- tion at once. This constraint automatically satisfies the

"exclusion principle.

" _If

_{there are}

l„or

^{fewer particles on}

a 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 of

j

^{particles moving to}

the right when the occupation number is k is the same as

(/„— j)

particles moving

to

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 (this}

was 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)!

⁴

^x 10,

^so these events are extremely rare (the simulations represent 21000sam- ples of 4000sites over 25000time steps, so the expected total number

of

such events is less than

10).

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

=

10

4).

Moreover, the universality class for the scaling properties is not expected

to

depend on such events, as the reactions take place in the zone where the density is low. These results may therefore justifiably described as equivalent

to

those reported in

[16].

^The

FQRTRAN 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 a

B

particle, it is important that the reaction takes account of parti- cles

of

difFerent types crossing over each other

(i.e.

, an A particle

at

sitei hopping to

i+

¹

at

the same time that a H particle at

i+1

^hops

^to

^site

i).

The reaction algorithm first removes such particles and then annihilates any re- maining Aand

B

particles occupying the same site. This isillustrated in

Fig. 1.

Four sites are shown, with initially two sites occupied by A particles (represented above the line, labeled 1—5) and two occupied by

R

particles (be- neath the line, labeled 6—

10).

In the difFusion step, each particles moves onto

a

neighbor

at

random, producing the state

(ii).

The reaction first deletes the A

— _R

_pair

REFINED 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)^from

state (ii),

^{leading to a state where there are}

B

particles

to

the left ofAparticles.

B.

Simulation results

the values they would have for an infinite system. Figure

2shows aplot

of

(a

—

_{b) as a function}

_of

_{(x/ti~ ) for three}

time values, displaying excellent rescaling. Asimpler test

of

finite-size effects is

to

show that the

total C

particle number

Kc = j ^C

^{dx is} proportional to

t

^~ . The inset

to Fig.

2 confirms that

Z~t

^~ is indeed independent

of

time.

Figure 3is a log-log plot of

X,

,

m,

, and

l,

as a function of time, where

X«) =

C(x, t)

^dx

)

(10)

where

C— : ^{j Bdt}

^{is the profile} of the reaction product.

This quantity differs from

x(q),

^{but since}

j ^Cdx

^{oc ti~2}

and 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 deviations

of

the particle profiles from An approximation to a Poissonian initial state

of

^av- erage density unity was prepared by performing 16 at- tempts to add an A particle, with probability

1/16,

^to each

of

the first 2000 sites of a 4000-site

lattice.

The other half of the lattice was similarly populated with

B

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 allowed

to

reenter the system at the end sites. The average density at the extremities was thus kept

at

the value unity.

In order

to

mimic the simulations in [16]^{as closely} as possible, instantaneous measurements were made

of l~~,

m, ^and the concentration profiles of the product and reagents

at

times 1000, 2500, 5000, 7500,.

. .

,25

000.

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 9

3.0 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 of

I,

,m, , and

X,

^{(see the text)}

from the MC simulations with Poisson initial conditions.

and

(~,

p~, and ^A~ are constants

that

^will be defined later. The straight lines are fits

to

the last eight points for

X,

⁽²⁾,

m,

⁽²⁾, and

l,

(i). The gradients for least-squares Gts

to

the curves in

Fig.

3 are listed in Table

I.

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 describing

m(t)

appears to be close

to 0.

29,instead

of 0.

375as they found. Second, the exponents describing X(q) appear

to

be independent of q. This means that

C(x, t),

^and

4058 STEPHEN

J.

CORNELL

TABLE

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 0.367

theoretical 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) ⁱⁿ

Fig. 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 is

too

narrow for conclusive deductions to be made.

Figures 5, 6,^and 7 are plots of

C, Pt(l~~),

^and

P _(m),

respectively, as a function of appropriate scaling vari- ables,

to

show the subjective quality

of

scaling for these quantities. The profiles

of 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)

=

_exp

mp K mp (14)

Using these values

of

_pq and Aq in

Eqs. (11)

^—

(13),

^one

would expect

m,

and

I, to

be independent of_q if the forms

(14)

^and

(15)

^{are valid. The} coincidence of the curves in

Fig. 3

confirms this.

0.40

g

^0.35

C

~

^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 for

I, ^m,

^and

^X

^(seethe

text) from the MC simulations.

FIG.

5. Scaling plot for

C(x, t),

^{forthe MC} simulations.

REFINED SIMULATIONS OF THE REACTION FRONT

FOR. . .

4059 0.8

0.0

~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

-2

C46

0

_-4

C4

I

aa-6

0

MC

'0jrp

. . .

(

i~0,g

:~. R

and

P

-&nglit_V

1 P1,MC

1 P1,PCAP

1 P1,PCAF

P,

^MC

P,

^PCAP

P,

^PCAF

R, PCAP R,PCAF RM&

FIG.

6. Scaling plot for Pt(I), ^{for the} MC simulations (MC) ^and the PCA simulations with Poisson (PCAP) ^and

full (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( )l

Pi(l)]

^against (m/m(

))

and (l/l(

)),

^re-

spectively, at

t =

25

000.

The

Y

ordinate has been shifted sothat all curves are coincident

at

the origin. The curve labeled RM~ is

logio[C(x,

²⁵⁰⁰⁰⁾

— _C(x,

₂₂

_500)],

which is approximately proportional

to R(x,

²⁵

000),

as

a

function

of (x/I( ))

. The straight line for this curve suggests that the reaction profile

R(x, t)

^{is also}

a

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 ^lead

to A,

being independent

of

_q without assuming a form for x('i)(t) ^{for all}

t,

so the values of (~ ^{used in}

Eq. (11)

^were chosen numerically in an ad hoc fashion.

III. PROBABILISTIC CELLULAR

AUTOMATA

MODEL

A. Description of

model

0.5

o~[Q^x

)

~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 up

to

two particles ofeach species at each site, labeled by the direction from which they moved

10 20 30

2 2 2

l

orm orx

FIG.

8. Fits of

P,

^{Pt, and}

^R

^{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 consists

of

changing the velocities

of

these particles and then moving the particles onto the neighboring sites ac- cording

to

their new velocities.

If

there are two particles per site, they both move in opposite directions, ^whereas

a

single particle will change direction with probability

p.

The value used in these simulations was p

=

_{2, so}

that

the particle forgets its previous velocity

at

each time step and the model is equivalent

to

the MCmodel with _lz

— — _1.

The reaction step consists ofchecking each site for si- multaneous occupancy

of

A and

B

particles

at

the start

of

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 an

A-B

pair

if

the A ar- rived from the left and the

B

^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 implementation

of

the algorithm simu- lates 64 independent systems

at

once.

The quantities

P _(m),

P&(l), and

R(x, t) at

^measure- ment time

t

were estimated by averaging over the interval

t(1 —

_b)

( ^t ( ^{t(1 +}

^{b), with b}

^{= 0. 05.}

^{We may estimate}

the order

of

magnitude

of

the systematic error that this introduces into the measured shape

of

these quantities.

Let

F _{(x, t)}

be the estimate

of

afunction I"

(x, t)

using the above method. Then

0.0

~PCAF,t=800

~PCAF,t=6400 x PCAF,t=51200

)KPCAF,t=409600

I

FIG.

7. Scaling plot for

P

_(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, _(,

_~)

(

^{' Ot}

1,

^{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 therefore

of

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, so}

^E

^{has the}

same scaling properties as

F.

Inorder

to

maximize the statistics, the reaction profile A was measured

at

every time step ^between

t(l ^—

h) and

t(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 implementation

of

this algorithm performed

3.

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 form

FIG. 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 results

R(T, t) =

exp

—

( —

)

⁽²⁵⁾

1. Poi

saon initial condition

x+

() —

_{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 be

too

close to the curves for

x,

. The straight lines are afit

to

the last five points, forthe lowest values ofq. The gradients ofleast-square fits for all the curves are summarized in Table

I.

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 each

of

the appropriate site variables (A particles for

x (

^0,

^B

^{particles for}

^x )

⁰⁾

with probability

4.

The lattice size was 4000 sites and at the boundaries particles were free

to

leave the system, with the density at the boundary maintained

at

an aver- age value of ₂ by allowing A particles to enter from the left and

B

particles

to

enter from the right, randomly with probability ^—,

.

Measurements were taken

at

times 200—102 400time steps, with the interval between mea- surements doubling progressively. The quantities

P _(m), P

_(l)and

B(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 ^power

of

the normalized qth moment ofthese quantities.

Figure 9shows a plot

of (a —

_{6) asa function}

_of

_(x/ti~2)

for three time values and

a

plot

of Z~t

^~ (where

E~ =

f ^Bdx)

^as

^a

^{function of}

^t.

^{These plots show}

^{that, just}

^as

in the MC simulations, there are no finite-size effects.

Figure 10is a log-log plot

of x,

,

m~,

and

l,

as a function of time, where

~2.

0

1.5

CO

1.0

~

bQ^O

05

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 of

l,

, m,^{~}, and}

x,

(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 in

Fig. 10.

The curves are much less noisy than those in

Fig.

4,by virtue

of

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~ ~} appears

to

increase initially, but the last few points appear also

to

decrease and in any case an asymptotic value

0.

375is ruled out.

The rescaled forms

of Pt(l), P

(m), ^and

B

are de- noted by

P

CAP in Figs. 6, 7, ^and 12 respectively.

Figure 8 shows

a

fit of

Pt(l), P

(m), ^and

R

to the forms

(14), (15),

^and

(25).

^Prom

Eq. (19),

^using

E(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).

^The

51 REFINED SIMULATIONS OF THE REACTION FRONT

FOR. . .

4061 0.32

~

0.30

0

~

^0.28

~W

~ 0

^0.26

om(2)

~^X(2)

0)(I)

C) p

Ei

0.24

0.0 0.2 0.4

I/log 10(t)

FIG. 11.

Effective exponents for

I, ^m,

^{and x (see}

the 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

0.2

2 4 5 6

log

_10(t)

FIG.

13.Log-log plot of

I,

,m, , and

x,

(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 condition8

Because

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 up

to x =

0and filled with

B

particles for

x )

^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- ble

to

perform simulations on a system that is effectively infinite, thus having no finite-size effects.

Simulations

of

64 000independent evolutions of

a

full lattice were run for 409 600time steps. Measurements

of

P _(m),

_P~(l), and

B

were made using the same method as for the Poisson initial condition. Results for these simulations are shown in Figs.

13

and

14. It

might be expected (considering the arguments in

[16])

^{that this}

case would be in a different universality class &om the case with randomness in the initial

state.

However, the results for the exponents (see Table

I)

^{are very close to} those measured for the case

of

Poisson initial conditions and the marked decrease of the exponents for x(2) (ar- guably towards

0.

25) is also seen in

Fig. 14. It

is inter- esting

to

note that the transient trends in ^{m~ ~} and ^{l~ ~} are in the opposite sense

to

the Poisson case.

Scaling plots for P~,

P,

^and

^B

^{are shown in Figs. 6,}

7,^and 12,denoted by

PCAF.

Plots of/ P~,

P,

^and

^B

may be found in

Fig.

8,^confirming

that

the profiles again have the forms

(14), (15),

^and

(25).

Figure 15is

a

plot of^{m~ ~} and ^{x~ ~} as

a

^function

of

the time-dependent current

Z~ = f ^Rdx,

^{from the simula-}

tions both with

(P)

and without

(F)

^Poisson fluctuations in the initial

state.

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 sensitive

to

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

~

^0.31

0

(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 for

I, ^m,

^{and x (seethe}

text) from the PCA simulations with full initial conditions.

4062 STEPHEN

J.

CORNELL 1.8

0.2

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,

exp

I— ^(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

IN

THE INITIAL STATE

The measured value m

t

^~ in Ref. [16]^was justified by an argument about the Poisson Huctuations in the initial condition. The argument went as follows: after time

t,

particles within a distance

t

^~ have had a chance of participating in the reaction. The number of particles within a distance

t

^/ is of order

t

^~

+ ct

^/ Since each reaction event "kills" precisely one A and one

B,

there is therefore a local surplus

t

^~

of

one of the species. The majority species therefore invades the minority species by a distance m, such that the number

of

minority particles between the origin and m is Since the particle profiles vary like

x/t ~,

this means that_In

j ^(x/t

^{~ )dx}

^t

^{~ )so m}

^{~ t}

^~

order

to

assess the validity of this argument,

it

is possible

to

apply it to

a

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 series

of

negative Dirac bpeaks for

x

&0 and positive Dirac b peaks for x

) ^{0. That}

^is,

f

^{p(z o)}

^{( z'i ( 2zxpi}

Suppose that xo

t,

^where

^a

^isexpected

^to

^{be less that}

i,

and let ^e

=

^ts+li~2l, with 0

(

^b

(

^(—

^—a).

^{Then the}

contribution to the integral in

(33)

^for ~x~

)

^{e is oforder}

exp(

—

e /t) exp(

— _{t ),}

which vanishes as

t

⁺oo. How- ever, for ~x~

(

^e, the argument

of

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 given}

by

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 ~ dx}

j

^{p(y, o)e "~ dy}

xp

(f ^xpjx,

^OIe

^'

^dT

^gyp(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 all

t )

^{0, p is zero}

^at

precisely one point, say

xp(t).

It

is possible

to

find the probability distribution of these zeros

P(xp)

^{over the ensemble}

of

initial states. The position xp is defined by p(xp,

t) =

0or, equivalently,

p(x

t=O) =)

_i=1

^{b(x-x) —} )

_i=1

^8(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 the}

^x,

^and

y; 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).

Then

j ^xp(x,

^{0) exp( x}

^/t)

^dx

^{— = t+} j ^x~(x)

^exp(

^—

^{x2/t) dx, the}

second term being typically much smaller than the G.

rst.

To 6.nd the leading contribution to xo, it ^{is sufficient}

to

replace

j ^xp(x,

^{0) exp(}

^—

^{x /t)dx in the} denominator by

t.

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 form

CK) CX)

(*p)

=

4

(p(x

o) p(y

o))

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 of

P(xp)

^is

of

the form

2n

(xp") =

^— dxg

. _dx2„exp ~—

x(p(zi,

0) ^'

p(z2„, 0)) + ^{. .}

(40) (2

)!

^{& t&}

^"~

2-.

^~~ 8

+

^0~~

(41)

For a distribution

of

the form

P(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 from

Eq. (30),

compared with the asymptotic pre- dictions of

Eq. (41).

From

Eq. (1),

^{the density difference}

(a —

_{b) in the}

reaction-difFusion problem, averaged over evolutions, obeys asimple difFusion equation. The quantity _p with the initial condition (26) ^is therefore equal

to (a —

_{b) for}

the initial condition with Poisson fluctuations used in the numerical simulations, with negative peaks correspond- ing

to

A particles and positive peaks corresponding

to 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 ^m

t

^~ may be applied equally ^well

to xo.

The reaction center shifts

to

compensate for a local majority oforder

t

^/ in one

of

the species and the argument predicts xo

t

^~

. It

is interesting

to

note that the correct exponent is obtained if the initial value

a(x,

0)

=

_{ap is} used instead of the value

a(z, t)

^oc

z/t

^~

at

time

t

in the balance equation

f ^{' 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 independently

of

the presence

of

an exclu- sion principle and ofrandomness in the initial

state.

The motion

of

the reaction center due

to

the Poisson noise ap- pears only

to

account fora contribution

of

order

t

^~

to

the reaction width, which isnot large enough

to

alter the scaling behavior. The measured exponent

= 0.

29 describ- ingboth the reaction width and the midpoint fluctuations appears in fact

to

be decreasing slowly in time, ^{with fa-} vorable evidence for an asymptotic value

0.

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 width

tp to the diffusion length

(Dt)

^{~~2 was} never smaller than

— _0.

_{2, whereas} the application

of

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 appear

to

be practical

at

present.

An investigation

of

the simulation procedure used by Araujo et ^a/. has revealed

a

few errors in the results pub- lished in

[16].

^A repeat

of

their simulations appears

to

confirm the results

of

the present article for

P

and P~

and the behavior m

t,

^{but does not find}

^{that 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 for

R ₍

exp[

—

_(z/m)s~2]) do not agree with the results

of

simulations. The approximation scheme they used would therefore not appear

to

be valid, unless it describes aregime inaccessible

to

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 as

t

^~ is far from conclusive and so needs to be _{put on}

a

sound theoretical basis either by an exact calculation or by

a

rigorous justification for the analogy with the static case.

ACKNOWLEDGMENTS

I

would like

to

thank Michel Droz, Hernan Larralde, John Cardy, and

Ben

Lee for many interesting discussions and Michel Droz for acareful reading

of

this manuscript.

I

would also like

to

thank Mariela Araujo for making details

of

her simulations available

to

me.

STEPHEN

J.

CORNELL [1]

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