Inhomogeneous domain-Inhomogeneous domain-structure of pair contact structure of pair contact

process withprocess with

diffusiondiffusion

Sungchul Kwon and Yup KimSungchul Kwon and Yup KimKyung Hee UniversityKyung Hee University

I. ContentsI. Contents

1. The inhomogeneous domain structure and its e1. The inhomogeneous domain structure and its effect on critical exponentsffect on critical exponents

- Unidirectionally coupled two-level systemsUnidirectionally coupled two-level systems2. Pair contact process with diffusion (PCPD)2. Pair contact process with diffusion (PCPD)- The inhomogeneous domain structure of PCPD- The inhomogeneous domain structure of PCPD- The effect of the domain structure on dynamic e- The effect of the domain structure on dynamic e

xponent, Z of PCPDxponent, Z of PCPD3. Summary3. Summary

1. Unidirectionally coupled PC→DP system1. Unidirectionally coupled PC→DP system (1) Model : S. Kwon & G. M. Sch(1) Model : S. Kwon & G. M. Schüü tz (2005)tz (2005) Consider two-level hierarchy. Consider two-level hierarchy.

PC = Parity conserving classPC = Parity conserving class DP = Directed percolation classDP = Directed percolation class

The faster decay and slower spreading of PC process thanThe faster decay and slower spreading of PC process than DP process DP process BB-level still exhibits DP critical behavior-level still exhibits DP critical behavior..

I. The inhomogeneous domain structure of unidirectionallI. The inhomogeneous domain structure of unidirectionally coupled systemsy coupled systems

A

A

pwithAA

pwithA

13

,02

B

B

pwithBB

pwithB

12

,0

withBAA

A

B

(2) The inhomogeneous structure at multicritical point(2) The inhomogeneous structure at multicritical pointAt multicriticality ( ), both levels are critical at the same time.At multicriticality ( ), both levels are critical at the same time.Initial condition : 2Initial condition : 2AA on on AA-level-level

The coupled area of size RThe coupled area of size Rcoupledcoupled : :

The spreading range of The spreading range of AA particles particles on on AA-level.-level.AA and and BB particles are unidirectionally particles are unidirectionally coupled in the coupled area !coupled in the coupled area !

The uncoupled area of sizeThe uncoupled area of size RRuncoupleduncoupled

Only B particles exist and Only B particles exist and the coupling is lostthe coupling is lost..

t

totalR

uncoupledR

coupledR

x

)()()( tRtRtR coupledtotaluncoupled

CB

CA pp ,

PC

DP

(3) Critical spreading of (3) Critical spreading of BB level at multicritical point level at multicritical point

Prediction :Prediction :

Simulation results :Simulation results :

The underestimate of zThe underestimate of zBB results from the double structure ! results from the double structure !

ZztRtRRtRR UBA zuncoupled

ztotalB

zcoupledA /2;~,~,~ 22222

265.1

,15.1

DPUB

PCA

zzz

zz

DPuncoupled

DPB

A

zz

zz

z

)2(26.1

)2(22.1

,)1(15.1

1.0~),1(~ PCDPz

uncoupledAB zztatRRR DP

II. Pair contact process with diffusion (PCPD)II. Pair contact process with diffusion (PCPD)

1.Basic model1.Basic model Single species with hard-core interactionSingle species with hard-core interaction

(1) Diffusion of particles : (1) Diffusion of particles :

(2) Reaction dynamics of pairs :(2) Reaction dynamics of pairs :

(3) Coupling dynamics between pairs and solitary (3) Coupling dynamics between pairs and solitary particles :particles :

00AA0 = 0 = SS = solitary or isolated particle , = solitary or isolated particle , AAAA = = PP = a pair = a pair

S SS S

P P P P

S SS S

DwithAA 00

2/)1)(1(0,0

,)1(0

DpwithAAAAAAA

DpwithAA

PSS SSP

particlessolitaryofdiffusionbycouplingCyclic

2. Critical behavior of PCPD2. Critical behavior of PCPDDefinition of various critical exponents Definition of various critical exponents

PCPD dynamical exponents :PCPD dynamical exponents :

Continuoulsy varying exponents with diffusion rate.Continuoulsy varying exponents with diffusion rate.However, the estimates are still controversial due to However, the estimates are still controversial due to Inaccuracy of critical pointInaccuracy of critical point Very slow approach to the scaling regimeVery slow approach to the scaling regime Unknown correction to the scaling or generic feature ?Unknown correction to the scaling or generic feature ?

Question : Question : Are there unknown corrections to the scaling ?Are there unknown corrections to the scaling ? One answer for Z is the inhomogeneous structure One answer for Z is the inhomogeneous structure

/,;~||~,||~,~ |||||| Zpp c

Z

tPtRtNtAt z ~,~,~,~,0 2

48.015.0,13.0

,0.217.1,28.017.0

Z

3. The inhomogeneous domain structure of PCPD3. The inhomogeneous domain structure of PCPDDue to the coupling Due to the coupling is expected in average. is expected in average. Then, we haveThen, we have

as in unidirectional coupling,as in unidirectional coupling,

RRUU = the size of uncoupled area in which = the size of uncoupled area in which

only solitary particles exist.only solitary particles exist.

If If RRUU exist, it can play the role of correction exist, it can play the role of correction

to the scaling of to the scaling of RRtotaltotal . .

Underestimate of Underestimate of zz of of RRtotaltotal

SorSSP

PS RR

t

x

UR PR

SR

UPStotal RRRR

PtotalU RRR

S

P

4. Monte Carlo simulation4. Monte Carlo simulation(1) Model (1) Model Soft-constrained PCPD : Phys. Rev. Lett. 90, 125701.Soft-constrained PCPD : Phys. Rev. Lett. 90, 125701. Bosonic PCPD with parallel update of all particles as followsBosonic PCPD with parallel update of all particles as follows

(i) Diffusion to n.n sites of all particles with rate D(i) Diffusion to n.n sites of all particles with rate D(ii) On-site reaction (DP process) : (ii) On-site reaction (DP process) : At each occupied site with ,At each occupied site with ,

[N[NA A /2] = The number of pairs at a site./2] = The number of pairs at a site.Criticality point = 0.795417(2), Criticality point = 0.795417(2), Physica A 361, 457 (2006).

]2/[

]2/[

102

32A

A

N

N

pwithA

pwithAA

1AN

(2) Simulation results(2) Simulation resultsIniti. Condi. = a pair, # of samples=10Initi. Condi. = a pair, # of samples=1055 , time = 5X10 , time = 5X1077

(A) The number of particles : averaged over all samples(A) The number of particles : averaged over all samplesPS NN ,

)1(33.0 PS 3137.0DP>

(B) Squared spreading distance(B) Squared spreading distance

0&0~

,1~

,0~

,0~

2

2

2

2

SPz

uncoupled

totalz

total

Sz

S

Pz

P

NNfortR

NfortR

NfortR

NfortR

U

S

P

)1(26.1),1(24.1),2(19.1 PSU zzzz

07.0)1(~ UP

zUPStotal zzwithAttRRRR P

265.1DPz

We cannot exclude DP type spreading behavior of PCPD

(C) Other exponents :

From the hyperscaling relation

We predict

)3(128.0

PPPP z 2/

20.0)18(172.016.0 pDP

III. SummaryIII. Summary

1. PCPD exhibits the inhomogeneous domain structure as in1. PCPD exhibits the inhomogeneous domain structure as in unidirectionally coupled systems.unidirectionally coupled systems.

2. Since the uncoupled area spreads more slowly than the coupled 2. Since the uncoupled area spreads more slowly than the coupled area does, it plays the role of the correction to the scalingarea does, it plays the role of the correction to the scaling of the total spreading distance.of the total spreading distance.

3. Taking the domain structure into account, we numerically confirm3. Taking the domain structure into account, we numerically confirm that the critical spreading is very close to that of DP class at thethat the critical spreading is very close to that of DP class at the given critical point of the soft-constrained PCPD.given critical point of the soft-constrained PCPD.

4. However, if the given critical point is not precisely measured, then4. However, if the given critical point is not precisely measured, then our estimates should be modified. But the domain structure is still our estimates should be modified. But the domain structure is still unchanged. So we still have the correction to the scaling which unchanged. So we still have the correction to the scaling which screen off the true scaling of Rscreen off the true scaling of R