Pair Contact Process with Diffusion (PCPD)

Niels Eweg

July 7, 2006

Abstract

The Pair Contact Process with Diffusion (PCPD) is an out-of- equi-librium reaction diffusion model showing a phase transition betweenan absorbing and an active state. We study this model at a constantdensity and we use an algorithm which exploits the fact that the den-sity in this model is very low. With this method, we find that therelation between density ρ and effective annihilation rate p is given by(p − pc) ∝ ρκ2 with κ2 = 2.3 ± 0.1. Finite-size effects are also stud-ied, giving (p − pc) ∝ L−κ1 with κ1 = 1.00 ± 0.05. The same methodapplied to DP gives κ2 = 2.7± 0.2 and κ1 = 0.95± 0.05. Since one ex-ponent agrees within error margins and the other within two standarddeviations, we cannot rule out that DP and PCPD belong to the sameuniversality class.

1

Contents

1 Theory (Model) 31.1 Non-equilibrium statistical models . . . . . . . . . . . . . . . 31.2 The Contact Process . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Critical Point p = pc . . . . . . . . . . . . . . . . . . . . 41.4 Characteristic exponents . . . . . . . . . . . . . . . . . . . . . 51.5 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Universality classes . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theory (Simulation) 92.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . 92.2 The Doubly Linked List . . . . . . . . . . . . . . . . . . . . . 102.3 Three Doubly Linked Lists . . . . . . . . . . . . . . . . . . . 112.4 Simulation with the 3-DLL Algorithm . . . . . . . . . . . . . 132.5 Simulation with constant ρ . . . . . . . . . . . . . . . . . . . 132.6 Calculation of peff . . . . . . . . . . . . . . . . . . . . . . . . 14

3 The Experiment 153.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 simulating DP . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 DP Model: Calculation of pc and critical exponents . . . . . . 183.5 Results of the DP-Simulation . . . . . . . . . . . . . . . . . . 223.6 simulating PCPD . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Conclusion 27

2

1 Theory (Model)

1.1 Non-equilibrium statistical models

To become familiar with the world of nonequilibrium statistical mechanicswe start to examine the Contact Process (CP). The Contact Process wasproposed by T.E. Harris in 1974 as a model of the spreading of an epidemic[1]. This is a logical starting point because it is one of the simplest formsof a nonequilibrium problem and also it will serve as a test model when weare going to use it to validate our future algorithm.

After the introduction of the Contact Process we will start to explainthe appearance of an absorbing state. We will examine the phase diagramand the different sorts of behaviour of the system. We will introduce a socalled critical point and then try to understand the critical behaviour of thesystem close to the critical point, which will result in the definition of anumber of critical parameters which determine the universality class of thesystem.

1.2 The Contact Process

In the original CP model each site of a lattice represents an organism thatexists in two states: either healthy or infected. In this terminology infectedsites are said to be occupied and healthy sites vacant. The infection spreadsthrough nearest-neighbour contact. An infected site can pass the disease toits neighbour at rate p/2. Infected sites can recover at rate d which is set tobe 1. This is shown in Figure 1.

w g w

w w w?

p

w g g

w w g?

p/2

g g w

g w w?

p/2

w

g?

1

Figure 1: The Contact Process

Because a site must have an infected neighbour to become infected it isimpossible for the system to ’escape’ from the disease-free state; this state istherefore called absorbing. The persistence of the epidemic depends on theinfection rate p. If p is sufficiently large the infection will spread throughoutthe whole system. The boundary between the spreading and the extinctionis marked by a critical point with p = pc.

3

time

t

position i

pc pc cpp>p=p<

Figure 2: The Contact Process with different values of p. Top row: randominitial conditions. Bottom row: a single infected seed.

1.3 The Critical Point p = pc

To show this dependence on the value of p figure 2 [8] shows the differentscenarios for different values of p (p < pc, p = pc, p > pc) from randominitial conditions (top) and from a single seed (bottom).

For the existence of a phase transition it is necessary that [7]:

a) at least one absorbing state is dynamically accessible,

b) there are two competing processes for particle creation and removal,

c) there is a mechanism which prevents the particle density from diverg-ing.

It has been proven [2, 3] that for the contact process at the critical pointpc a phase transition exists and that this phase transition is continuous.Starting from a finite population the density dies out exponentially for p < pc

while for p > pc the radius of the populated area grows ∝ t. In theory anactive steady state at p = pc only exists in the infinite-size limit of thesystem since any finite system will eventually be trapped in the absorbingstate. In simulation, although we are working with a system of finite size,this quasi-stationary state exists for a long enough time for us to be able tostudy it.

4

1.4 Characteristic exponents

In equilibrium statistical mechanics phase transitions are characterized byscaling laws, which describe the behaviour of certain quantities close thecritical point. If we introduce the stationary density ρ in the active station-ary state and the distance to the critical point ∆ ≡ p − pc then we maydefine the order-parameter exponent β as:

ρ ∝ ∆β. (∆ > 0) (1)

In a double-logarithmic plot of ρ versus ∆, the exponent β is the slopeof the straight line. There are no exact results for pc but simulations [4] givepc

∼= 3.2978 while β ∼= 0.277.

If we look at the phase diagram [4] (figure 3) we see that pc can only bereached in the limit ρ → 0.

Figure 3: Phase Diagram of the Contact Process

The parameter β is called a critical exponent. Besides β which is associ-ated with the particle density, two other exponents are defined characteriz-ing the independent correlation lengths: the spacial length scale ξ⊥ and thetemporal length scale ξ‖. Near the phase transition the following relationshold:

ξ⊥ ∝ ∆−ν⊥, (∆ > 0) (2)

ξ‖ ∝ ∆−ν‖ , (∆ > 0) (3)

5

with ν⊥ and ν‖ the corresponding critical exponents. The physical meaningof these exponents is explained in figure 4.

ξ

||ξ

||ξξξ ||

ξ

D

||

C

x

p>pcp<pc p>pcp<pc

ξ

A B

t

Figure 4: Interpretation of the correlation lengths ξ⊥ and ξ‖ in an almostcritical DP process below and above criticality. In panels A and B a clusteris grown from a single active seed while in panel C a fully occupied latticeis used as initial state. Panel D shows a stationary DP process in the activephase. The indicated length scales ξ⊥ and ξ‖ must be interpreted as averagesover many independent realizations.

Different physical systems can be categorized into different sorts of classes,called universality classes. If two systems share all their critical exponents,they are called to belong to the same universality class.CP is not a universality class by itself but it belongs to a more general classcalled the Directed Percolation (DP) class, which is the largest and bestknown class connected with absorbing state transitions.

1.5 Other models

Some other models in statistical mechanics are the Parity Conserving (PC)process, the Pair Contact Process with Diffusion (PCPD) and the TripletContact Process with Diffusion (TCPD). The characteristics of these modelsare summarized in figure 5.

1.6 Universality classes

The question to which universality class a model belongs is in many caseseasy to answer: the majority of models belongs to the Directed Percolationclass. A second category is formed by models like PC, which form a differ-ent class. This is believed to be related to an extra form of symmetry theypossess (in this case mod2 symmetry). For PCPD and TCPD the questionto which universality class they belong is not so easy to answer. The classi-fication has been an open problem in non-equilibrium statistical mechanicsfor a while now and a lot of speculation has been going on about whether

6

The Parity Conserving Process (PC):

0A0 → AAA with rate (1 − p), (4)

AA → 00 with rate p .

The Pair Contact Process with Diffusion (PCPD):

AA0 → AAA0AA → AAA

with rate(1 − p)(1 − d)

2, (5)

AA → 00 with rate p (1 − d) ,

A0 ↔ 0A with rate d .

The Triplet Contact Process with Diffusion (TCPD):

AAA0 → AAAA0AAA → AAAA

with rate p/2 , (6)

AAA → AA0AAA → 0AA

with rate (1 − p)/2 ,

A0 ↔ 0A with rate (1 − p)/2 .

Figure 5: Non-equilibrium processes in Statistical Mechanics

these models belong to an existing class or form new classes by themselves.

Kockelkoren and Chate propose the classification which is given in table 1.

Besides this classification a lot of other scenarios have been proposed.In their review article about PCPD (2002) [6] Henkel and Hinrichsen madea list of all the existing viewpoints on the classification of PCPD. Some ofthem are:

⊲ should represent a novel universality class with a unique setof critical exponents;

⊲ may represent two different universality classes dependingon the diffusion rate;

7

Table 1: Universality classes of different processes. m and n are the initialnumber of particles involved in respectively a creation or annihilation reac-tion. ∅ stands for a system with no non-trivial phase transition. The secondmarkings indicate the change of class whenever mod2 or mod3 conservationplays a role.

m/n 1 2 3

1 DP DP/PC DP/∅2 DP PCPD PCPD/∅3 DP DP TCPD4 DP DP DP

⊲ may cross over to DP after a very long time [10];

⊲ may have exponents depending continuously on the diffu-sion constant d.

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2 Theory (Simulation)

2.1 Monte Carlo Simulation

When simulating the models presented in chapter 1 we make use of MonteCarlo (MC) simulations. We will first introduce the most basic form ofthese simulations. In the next chapter we will introduce the doubly linkedlist (DLL) which will improve the efficiency of our algorithm significantly.

The most basic way of storing the particle configuration of the system isto make an array of length L (L is the size of the lattice) and then assigneither a 1 or a 0 to each entry when the site is occupied or vacant. Whatwe then do is schematically shown in figure 6.

Pick a site Pick a reaction Reaction possible? Perform reaction

?

- - -

6

yes

no

Figure 6: Basic Monte Carlo Simulation

We have to make sure that the performed reactions happen at the cor-rect rate. For the Pair Contact Process with Diffusion the probabilities toperform a reaction are given in figure 5. Because these probabilities do notadd up to 1 we multiply the probabilities with a factor PN

S , where PN isthe probability of picking a site (in this case 1/L) and with a scaling factorS = 2d+ p(1− d)+ (1− p)(1− d). For each MC-step the time is raised witha constant ∆T = 1

LS .

The rates for the different reactions will be:

Rdl = Rdr =PNd

S;

Ra =PNp(1 − d)

S;

Rpl = Rpr =PN (1 − p)(1 − d)

2S; (7)

with:

∑Ri ≡ 1 . (8)

9

In equation (7) the following abbreviations are used: ‘dl’ = diffuse left, ‘dr’ =diffuse right, ‘a’ = annihilate, ‘pl’ = procreate left and ‘pr’ = procreate right.

The algorithm with this basic MC-simulation is given in figure 7.

s=2d+p(1-d)+(1-p)(1-d) #scaling factor

i=random*L #select a particle

j=next(i)

k=next(j)

s=random*s

if (s<d) # j. -> .j

else if (s<2d) # .j -> j.

else if (s<2d+p(1-d)) # jk -> ..

else if (s<2d+p(1-d)+(1-p)(1-d)/2) # jk. -> jkl

else # .ij -> hij

time += P_N/s

Figure 7: Pseudo code for basic MC-simulation

The big disadvantage of this method is its low efficiency. Because the siteand type of reaction are randomly selected the chosen reaction can oftennot be performed and we have to start all over while nothing has changed.Especially at low densities, when most of the time empty sites are selected,only a small amount of choices is successful. A solution to this problem isthe application of the so called Doubly Linked List (DLL).

2.2 The Doubly Linked List

In the Doubly Linked List we do not store all the sites in our configurationbut we store only the occupied sites. Therefore the array can be of sizeN (with N the number of particles) instead of size L. Now whenever werandomly select a site we are sure there is a particle.

To implement this idea we introduce three separate arrays: prev[i]

which is a pointer to the previous particle, next[i] which is a pointer tothe next particle and gap[i] which contains the size of the gap betweenparticle i and the next particle. Figure 8 shows how a particle configurationis stored [9].

Notice that the absolute positions of the particles on the lattice are notstored. This is not necessary because the lattice is a closed ring. An extrameasure we have to take is updating the lattice after every reaction. If forexample we remove a particle somewhere from the array, an empty spot in

10

A00AA0AAA000A0

i 0 1 2 3 4 5 6 7 8 9

prev[i] 6 0 1 2 3 4 5

gap[i] 2 0 1 0 0 3 1

next[i] 1 2 3 4 5 6 0

Figure 8: example of a particle configuration with DDL

the array arises. We chose to always take the particle at the end of the arrayto fill the empty spot.

The reaction rates will stay the same as in equations (7), only for thisnew algorithm PN = 1/N which results in ∆T = 1/NS.

Also the algorithm given in figure 7 will not change, except for the siterelation i=random*L which becomes i=random*N.

This algorithm is a lot more efficient then the first one but improvementsare still possible. For example the choice of the reaction: a lot of randomlyselected reactions are not possible. In the next chapter we will fine-tune theDLL-algorithm so no impossible reactions will be chosen.

2.3 Three Doubly Linked Lists

To avoid randomly picking reactions which are not possible at the selectedsite we introduce the three linked list algorithm. In this algorithm we storethe particle configuration in three separate groups depending on the numberof neighbors the particle has (0, 1 or 2). Each group is of the same form asthe DLL presented in the previous chapter. Off course these three groupscan be stored in only one DLL if we keep track of the offsets where eachgroup starts. Figure 9 gives an example of the new storage method [9].

A00AA0AAA000A0

i 0 1 . . . 7 8 9 10 . . . 15 . . .

prev[i] 1 15 . . . 0 7 8 15 . . . 9 . . .

gap[i] 2 1 . . . 0 1 0 3 . . . 0 . . .

next[i] 7 0 . . . 8 9 15 1 . . . 10 . . .

Figure 9: example of a particle configuration with three DDL

If we now perform a reaction, not only do we have to fill the eventual

11

gaps in the array; also, because the number of neighbors may have changed,we have to make sure that the particles are stored in the correct group. Foreach group the number of particles (N#neighbours) is stored and updatedafter each reaction.

The advantage of this method is the fact that a reaction can only takeplace with a particle from a certain list. If such a particle is selected it iscertain the reaction is possible. Figure 10 gives the different reactions, thegroup of particles the reactions are possible for and the number of possiblesites in the complete system where the reaction can be performed.

Reaction: Lists: Possibilities

diffusion 0, 1 2N0 + N1

annihilation 1, 2 (N1 + 2N2)/2procreation 1 N1

Figure 10: Different reactions and the number of possible locations. Thefactor 2N0 arises because the particle can diffuse both to the left and to theright. The term (N1 + 2N2)/2 is the total number of pairs in the system.

We now choose the probabilities to be:

Pd = d(2N0 + N1)/S ;

Pa = p(1 − d)(N1 + 2N2)/(2S) ;

Pp = (1 − p)(1 − d)N1/(2S) ; (9)

with:

S = d(2N0 + N1) + p(1 − d)(N1 + 2N2)/2 + (1 − p)(1 − d)N1/2 . (10)

If we calculate the rates as Piπ, where π is the probability that a site isselected, we get the following reaction rates:

Rd =d(2N0 + N1)

S

1

2N0 + N1

=d

S;

Ra =p(1 − d)(N1 + 2N2)

2S

2

N1 + 2N2

=p(1 − d)

S;

Rp =(1 − p)(1 − d)N1

2S

1

N1

=(1 − p)(1 − d)

2S. (11)

12

So we see that if we take ∆T = S−1 we get the correct rates.

2.4 Simulation with the 3-DLL Algorithm

So far we have altered our algorithm in such a way that only relevant re-actions to every chosen site are picked. For most purposes this algorithmworks very well and systems with typical size L = 106 and initial density ofρ = 0.5 can be very well simulated within a decent amount of CPU-time upto total times of T = 106.

There are two main problems when using this algorithm in simulations:

• Starting from random initial conditions it takes a long time to reachthe steady-state-phase, especially close to the critical point.

• Because of the singular character of the system at the critical point,simulation time increases tremendously when ρ decreases.

Because no exact results on the cluster properties of this sort of systemsare known, it is not possible to theoretically construct an initial configura-tion that is already close to the steady state. Especially at very low densitiesthis initial period before the steady state is reached can be very long. Thereason for this is that at low densities we start with mainly single particlesand therefore only diffusion takes place before two or more particles ‘meet’.

For the second problem we propose the following: instead of a dynamicsystem which starts from a certain initial condition and develops accordingto the reactions in equations (6) we start with a system and keep the numberof particles constant; for every time a particle is created another particlemust be annihilated.

2.5 Simulation with constant ρ

First of all we have to ask ourselves if a simulation with this constant particlealgorithm is equivalent to our original one. For example, because for everyannihilation reaction there have to be two procreation reactions (2 particlesare annihilated in one reaction) the fraction of the rates of annihilation isnot what it should be according to (6):

RaRp

=p(1 − d)

(1 − p)(1 − d)/2=

2p

1 − p, (12)

13

but has become:

RaRp

=1

2. (13)

It turns out that although the rates have changed the two processes areequivalent, which is proven in ref. [11].

In simulations we have to make sure that the number of diffusion re-actions is still in the right proportion to the number of annihilation andprocreation reactions. Every time an annihilation reaction and two procre-ation reactions are being performed counts as three separate reactions, sothe probability Pd has to be a factor of three bigger than the one in theusual model. The probabilities now turn out to be:

Pd =3d(2N0 + N1)

S,

Pa+2*p =p(1 − d)(N1 + 2N2)

2S+

(1 − p)(1 − d)N1

2S, (14)

with:

S = 3d(2N0 + N1) + p(1 − d)(N1 + 2N2)/2 + (1 − p)(1 − d)N1/2 . (15)

2.6 Calculation of peff

In this model we do not use a fixed value of p but we start with an initialvalue pinit and then calculate the effective value peff after every reaction.Because of equation (13) we know that:

peff(1 − d)(N1 + 2N2)/2

(1 − peff)(1 − d)N1/2=

1

2, (16)

and therefore:

peff =N1

3N1 + 4N2

. (17)

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3 The Experiment

3.1 Outline

As mentioned before the overall goal of our experiments is to simulate thePCPD model with the algorithm from chapter 2.5 which makes use of adifferent sort of dynamics and therefore is not completely identical to theusual form of the model. Secondly we like to calculate the critical exponentsand the critical value pc and compare the universality classes of the DP andthe PCPD models.

So the goals we hope to establish are:

• Simulate the PCPD model with different sort of dynamics,

• more precise determination of the critical values for PCPD than cur-rent literature values,

• more precise determination of pc for PCPD than current literaturevalue,

• comparison of the universality classes of DP and PCPD.

In order to determine whether the method we will use is valid (although the-oretically it is justified [11]) we will first apply it to the DP model. Whenit is possible to reproduce the expected results for the DP model with thismethod we assume that also for PCPD this is valid.

So, what we will do is schematically shown in figure 11.

Figure 11: Method of determining the universality class of the PCPD model

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3.2 simulating DP

When simulating the DP model (with certain ρ, Tf and L) the typical resultwe get looks like figure 12.

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6

p_c

log(t)

Figure 12: Typical result of a simulation of the DP model (L = 4640000,N = 51040, ρ = 0.011).

We see that after a certain thermalisation time the system reaches a stablesituation (typically after 104 < t < 105). When the system reaches thisstate we can calculate pc (Fig. 13).

3.3 Data

In our simulations the value of pc still depends on two parameters: the sizeL and the density ρ of the system. We are mostly interested in the thermo-dynamic limit, with the limits ρ → 0 and N → ∞ (and consequently alsoL → ∞).

In our simulations we therefore simulate the model at a large number ofdifferent values for both the size of the system and the density, so we canextrapolate these results to, hopefully, find the correct value of pc.

Typically we simulate the model with:

0.001 < ρ < 0.4 , (18)

16

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6

p_c

log(t)

3.27

3.275

3.28

3.285

3.29

3.295

3.3

3.305

3.31

3.315

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

p_c

log(t)

Figure 13: Calculation of pc (L = 4640000, N = 51040, ρ = 0.011). pc =3.2943.

103 < L < 108 . (19)

Of course not all combinations of ρ and L are possible. We will have to havea substantial number of particles (ρL) to get valid results. Also systemswith a number of particles which is too large (ρL > 105) we do not simulatebecause of the extremely long simulation time.

17

When all simulations are performed we will estimate pc and the criticalexponents (see fig. 11).

3.4 DP Model: Calculation of pc and critical exponents

The calculations we will perform in this section are schematically shown infigure 14.

Figure 14: schematic diagram of the calculations

With the method presented in chapter 3.3 we calculate for every single sim-ulation we have performed the value pc. These values of pc still depend onboth L and ρ so this is the pc(ρ, L) in figure 14.

First we start with ordering the data in groups of data that share the samevalue of ρ. We have the following values of ρ:

ρ ∈ {0.001, 0.002, 0.003, 0.006, 0.011, 0.019, 0.034,

0.062, 0.111, 0.130, 0.150, 0.180, 0.200} (20)

For each data-set with the same ρ we take the limit L → ∞ and calculatethe corresponding exponent.

As an example we will now perform this calculation for one data-set withρ = 0.034.

The data we will use is plotted in figure 15.

The difficulty we encounter when trying to fit an exponential function throughthese points is the extremely large x-range in contrast to the very small y-range. In order to improve this we scale the figure in the following way:

L → L−1/10 , (21)

18

3.255

3.26

3.265

3.27

3.275

3.28

3.285

3.29

3.295

3.3

1000 10000 100000 1e+006 1e+007

p_c

L

Figure 15: Data of the DP model with ρ = 0.034

pc → q − pc [q ≃ 3.298] . (22)

After this scaling the data looks like figure 16.

When we fit an exponential function through these points we get the resultin figure 17.

The fit is made with the function:

y = λ1xκ1 . (23)

The bar denotes that these are the values in the scaled version of the data.Besides determining λ1 and κ1 we also find the correct value for q to makesure the fit is going through the origin. We are mainly interested in thevalues of κ1 and q. In this case they are:

κ1 = κ1/10 = 0.96 ± 0.03 , (24)

q = q = 3.2977 ± 0.0003 . (25)

19

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44

q-p_

c

L^(-1/10)

Figure 16: Data of the DP model with ρ = 0.034, scaled

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44

q-p_

c

L^(-1/10)

0.0001

0.001

0.01

0.1

0.1 1

Figure 17: DP model: best fit through data with ρ = 0.034.

When we have done this sort of calculation for all our sets with differentvalues of ρ we have one set κ1(ρ) and one set q(ρ). The q(ρ), we believe,should still have a ρ dependance but the κ1(ρ) however should actually beindependent of ρ.

20

q(ρ) and κ1(ρ) are plotted in figures 18 and 19.

3.296

3.298

3.3

3.302

3.304

3.306

3.308

3.31

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

p_c

rho

Figure 18: q(ρ) with best fit.

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

kapp

a_1

rho

Figure 19: κ1(ρ) with best fit.

The same as for the best fit in figure 17 we also obtain for figure 18 valuesfor, in this case, κ2 and pc:

21

κ2 = 2.7 ± 0.2 , (26)

pc = 3.2973 ± 0.0001 . (27)

For figure 19 things are a little bit more complicated. We expected thedata to be fairly constant. This seems to be the case but the error marginssometimes are too large to be absolutely sure. If we assume the data to beconstant we obtain a value for κ1 which is 0.95 ± 0.05.

We can check our calculation to plot all the data in one single figure andtry if, with our calculated values for pc, κ1 and κ2, we can create a so called’data-collapse’.

In figure 20 we have done this:

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

3.29

73 -

p_c

+ r

ho ^

2.7

(rho * L) ^ -0.95

Figure 20: ’Data Collapse’ of all data for the DP model.

Like we expected we obtain a straight line through the origin.

3.5 Results of the DP-Simulation

Our simulation of the DP-model so far has given the following information:

a value for pc:

22

pc = 3.2973 ± 0.0001 , (28)

and two exponents with the values:

κ1 = 0.95 ± 0.05 , (29)

κ2 = 2.7 ± 0.2 . (30)

Our first task now is to identify these exponents. Let’s start with κ1. Weexpect this exponent to be related with the critical exponent for the spatiallength scale (ν⊥) because we varied the value of L in the calculation. Fromequation (2) we expect:

(p − pc) = L−1/ν⊥ (31)

If we compare our calculated result with the literature value (ν⊥ =1.0968, 1/ν⊥ = 0.9117) we see that this value is slightly lower, but withinerror margins.

Likewise we also would like to identify κ2. One might think κ2 is relatedto β because we varied ρ when calculating this exponent. If we rewriteequation (1) we get:

(p − pc) = ρ1/β , (32)

so we expect

κ2 = 1/β . (33)

If we compare the literature value for 1/β (β = 0.27649, 1/β = 3.6167)with our calculated value for κ2 we see that these are, although of the sameorder, not identical.

Why is it that at least one of the values of these exponents differ fromthe exponents we expect to find from the DP model? The reason for thisprobably is that, as explained earlier, we use a model with different dynam-ics. For example we extracted the density-time correlation from the originalmodel. So instead of finding the exponents discussed in chapter 1.4 we findan other set of exponents defined by the new intrinsic properties of our newmodel. We expect that there is a relation between these new exponents and

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the original ones but it is not a trivial task to analyze this.

The last thing we have to do is to compare the calculated value for pc

with the literature value which is pc = 3.29785. We see that our value,error margins included, is slightly lower than the literature value. Wetherthis difference arises because our experiment is inaccurate or because theliterature value is incorrect we cannot say.

So to summarize the results: our model has resulted in values for pc

and ν⊥ with moderate precision. Because of this we can fairly say thatthe model we use is valid to perform simulations of these sorts of models.Unfortunately we have not been able to calculate all the critical exponentsbelonging to the original DP model.

3.6 simulating PCPD

Now we have done all the simulations for DP and analyzed them we areready to have a look at the model we are actually interested in: the PCPDmodel. Here we follow exactly the same way as described in chapter 3.2. Be-cause the details for the simulations are identical we will not describe themas detailed as we have done for the DP model but mainly present the results.

Like DP, for PCPD we also order the data in groups with the same value ofρ. We have the following values for ρ:

ρ ∈ {0.001, 0.002, 0.004, 0.005, 0.008, 0.010, 0.020,

0.040, 0.050, 0.080, 0.100, 0.200} (34)

For each value of ρ we now calculate q(ρ) and κ1(ρ). When we plot thesevalues it looks like figure 21 and 22.

We find the following values for κ1 and pc:

κ1 = 1.00 ± 0.05 , (35)

pc = 0.1526 ± 0.0001 . (36)

From figure 21 we find the following value for κ2:

κ2 = 2.3 ± 0.1 . (37)

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-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

q-p_

c

rho

1e-008

1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.001 0.01 0.1 1

q-p_

c

rho

Figure 21: q(ρ) with best fit.

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

kapp

a_1

rho

Figure 22: κ1(ρ) with best fit.

We see that the results of the simulation of the PCPD have a strong resem-blance with the results of the simulation of the DP model.

Again we check our calculations by plotting all data in one figure and see if

25

we can create a data-collapse with the calculated values for pc, κ1 and κ2.This is done in figure 23.

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

q-p_

c+rh

o^ka

ppa_

2

(rho*L)^-kappa_1

Figure 23: Data collapse of all data for the PCPD model.

Also this time we obtain a straight line through the origin, exactly what weexpect.

The value for pc we found is (like DP) very close to the literature value (pc =0.15245). If we compare the literature values for the exponents (1/ν⊥ = 0.71,1/β = 2.0) we see that they are not identical to our calculated exponentsbut of the same order and κ2 is even within two standard deviations of theliterature value.

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4 Conclusion

We simulated the PCPD model with a modified kind of dynamics. We wereinterested in whether this algorithm would be suited for computer simula-tion and whether the results obtained would be more precise than valuesfound in the literature.

What we found out is that for reaction diffusion models like DirectedPercolation and the Pair Contact Process with Diffusion the simulation canbe done faster than with traditional algorithms because of the fact that wemake use of a system with constant density. Disadvantage is that we werenot able to use the simulation to calculate all the critical exponents. Thevalue for pc on the other hand could be calculated with moderate precision.

Perhaps a field-theoretical analysis can show a relation between the ex-ponents we calculated and the ones related to the original model. Whensuch a relation can be found this algorithm might be used to calculate theexponents more precisely and perhaps clarify the question to which univer-sality class PCPD belongs.

Because the exponents we calculated for the DP model and the PCPDmodel coincide within two standard deviations we cannot rule out the pos-sibilty that both models belong to the same universality class.

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References

[1] T.E. Harris, “Contact Interactions on a Lattice”, Ann. Prob. 2, 969(1974).

[2] T.M. Liggett, Interacting Paricle Systems, (Springer, New York, 1985).

[3] R. Durrett, Lecture Notes on Particle Systems and Percolation,

(Wadsworth, Pacific Grove, 1988).

[4] J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice

Models, (Cambridge University Press, Cambridge, 1999).

[5] J. Kockelkoren and H. Chate, “Absorbing Phase Transitions ofBranching-Annihilating Random Walks”, Phys. Rev. Lett. 90, 12570(2003).

[6] M. Henkel and H. Hinrichsen, “The non-equilibrium phase transition ofthe pair-contact process with diffusion”, J. Phys. A Math. Gen 37,R117 (2004).

[7] K. Park, H.Hinrichsen and I. Kim, “Phase Transition in a TripletProcess”, Phys. Rev. E 66, 025101 (2002).

[8] H. Hinrichsen, “Nonequilibrium Critical Phenomena and PhaseTransitions into Absorbing States”, Adv. Phys. 49, 815 (2000).

[9] Figures taken from: T. van Leeuwen, “A doubly linked listimplementation for an MC simulation of the Pair Contact Process withDiffusion”, unpublished, (2004).

[10] G.T. Barkema and E. Carlon, “Universality in the pair contactprocess with diffusion”, Phys. Rev. E 68, 036113 (2003).

[11] H.J. Hilhorst and F. van Wijland, “Equivalence of stationary stateensembles”, Phys. Rev. E 65, 035103 (2002).

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