Universality classes of absorbing phase transitions in generic branching-annihilating particle systems

TL;DR

This paper investigates how bias in one-dimensional branching-annihilating particle systems affects their phase transition universality classes, revealing that bias preserves the directed percolation class but modifies others.

Contribution

It demonstrates that introducing bias does not change the DP universality class but causes the PCPD and PC classes to crossover or transform into new classes.

Findings

Bias does not alter the DP universality class.

PCPD class crosses over to DP with bias.

PC class changes to a new universality class under bias.

Abstract

We study absorbing phase transitions in systems of branching annihilating random walkers and pair contact process with diffusion on a one dimensional ring, where the walkers hop to their nearest neighbor with a bias $\epsilon$. For $\epsilon=0$, three universality classes: directed percolation (DP), parity conserving (PC) and pair contact process with diffusion (PCPD) are typically observed in such systems. We find that the introduction of $\epsilon$ does not change the DP universality class but alters the other two universality classes. For non-zero $\epsilon$, the PCPD class crosses over to DP and the PC class changes to a new universality class.

Tables

Table 1. Table 1: Numerical estimate of critical points and critical exponents in BARW and PCPD for various values of ϵ italic-ϵ \epsilon obtained using Monte carlo simulations. The numbers within parenthesis represent the error in the last place of decimal. The errors are determined from eye estimate in fitting the exponents in the power laws and the scaling function.

Model	$ϵ$	$p_c$	$\alpha$	$\beta$	$z$	$\theta$
BARW	0	0.1070(1)	0.161(1)	0.278(1)	1.58(1)	0.251(1)
m=1	0.1	0.08355(3)	0.159(2)	0.276(1)	1.59(1)	0.250(1)
	0.3	0.05819(4)	0.159(2)	0.275(2)	1.58(1)	0.251(2)
	0.5	0.04469(2)	0.158(1)	0.276(1)	1.58(1)	0.250(2)
BARW	0	0.7215(5)	0.284(3)	0.92(4)	1.75(1)	0.491(2)
m=4	0.1	0.5620(2)	0.232(4)	0.55(1)	1.72(1)	0.381(5)
	0.3	0.4182(2)	0.229(3)	0.53(1)	1.71(2)	0.378(3)
	0.5	0.3369(2)	0.224(5)	0.51(2)	1.74(2)	0.376(2)
		$q_c$
PCPD	0	0.13353(6)	0.221(3)	0.43(1)	1.72(2)	0.346(1)
	0.1	0.11898(4)	0.159(4)	0.28(1)	1.59(2)	0.254(2)
	0.3	0.10087(3)	0.159(1)	0.275(3)	1.59(1)	0.251(2)
	0.5	0.08982(4)	0.159(1)	0.275(2)	1.59(1)	0.251(1)

Equations

$$\rho (t)\sim t^{-\alpha }$$

$$\rho (t,L)\sim t^{-\alpha }f(t/L^z),$$

$${\rho }_a\sim (q_c-q{)}^{\beta }$$

$$C(r)\sim r^{-\theta }$$

Full text

I Introduction

Many reaction-diffusion systems show a second-order phase transition from a fluctuating active state to a non-fluctuating absorbing state as some control parameter is tuned hinrichsen_rev ; odor_rev ; hinrichsen_book . A wide range of models corresponding to phenomena such as epidemic spreading mollison_epidemic , catalytic chemical reactions chemical , transport in disordered media havlin_transport , forest fire albano_fire , biological evolution lipowski_evolution , surface roughening hinrichsen_rough ; roughening , self-activated biological structures berry_biology , etc show absorbing phase transitions. These transitions are classic examples of nonequilibrium phase transitions. Studying the critical behavior and universality classes of such transitions is extremely important from theoretical perspective and for understanding the phase transition in reaction diffusion systems.

A large number of absorbing phase transitions in nonequilibrium systems have been observed to belong to the directed percolation (DP) universality class. It has been conjectured by Janssen and GrassbergerDP_conjecture that continuous absorbing phase transitions in reaction-diffusion system with short-range interactions, characterized by a non-negative scalar order parameter and with no additional symmetries, conservation laws and quenched randomness belong generically to the DP universality class. The robustness of DP universality class has been a matter of great interest among researchers. The parity conserving (PC) universality class pc_class , the universality class of pair contact process with diffusion pcpd_class , the voter universality class voter_class and the Manna universality class in sandpile models manna_class are some noteworthy universality classes in nonequilibrium phase transitions whose critical behavior is different from that of DP.

In this work, we have focused on branching annihilating random walks (BARW) and pair contact process with diffusion (PCPD), where three universality classes, namely DP, PC and PCPD have been reported. In BARW, a diffusing random walker $A$ can branch to produce $m~{}(m>0)$ new off-springs $A\to(m+1)A$, or two of them can annihilate ($2A\to\O$) upon contact. The parity of the system which is defined as the total number of walkers modulo 2 is not conserved when $m$ is odd. Depending upon parity, the critical behavior in such systems is DP when $m$ is odd and PC when $m$ is even pc_class . PC class is also referred to as directed ising (DI) class because PC critical behavior can also be realized in spin systems when a spin-flip Glauber dynamics compete with spin-exchange Kawasaki dynamics menyhard_nikem . In PCPD pcpd_class , two diffusing walkers in contact only can produce new off-springs ($2A\to 3A$) or they can annihilate ($2A\to\O$). Unlike in PC, parity does not affect the critical behavior in PCPDparity_PCPD .

The crossover behavior between these universality classes have been extensively studied. It has been found that the PC class crosses over to DP by introducing a dynamics which breaks the modulo 2 conservation. This is done by producing both even and odd number of off-springs while branching park_park_PC_DP . In addition to PCPD dynamics, if unary branching and annihilation are introduced, then PCPD crosses over to DP when the unary process does not conserve parity pcpd_dp ; pcpd2_dp or to PC if it conserves parity pcpd2_dp . This suggests that parity and $n-$narity of the branching process plays a crucial role in determining the universality class of the transition. It is also found that diffusion, or its absence affects the critical behavior in a major way. For example, in absence of diffusion, parity conserving BARW models having spatially asymmetric branching can have additional conservation laws depending upon the initial conditions, and consequently, the decay exponent varies from the PC exponent hinrichsen_rough . The PCPD class crosses over continuously to DP when solitary diffusing walkers are annihilated upon contact with a certain probability which determines the value of the critical exponents noh_park . When diffusion of single walkers is completely forbidden in PCPD, although the system has multiple absorbing states, its critical behavior switches over to DP pcp_jensen . BARW has also been studied with Lev́y walkers. The additional long-range correlations that builds up in the system due to long-range interactions via Lev́y flights leads to continuous variation of critical exponents for both DP and PC universality classes albano_DP_conti ; vernon_PC_conti . The above mentioned perturbations either change the parity, or brings in additional conservation laws, long-range interactions, or arrests the diffusion dynamics.

We ask what happens to DP, PC and PCPD universal critical behaviors when only local perturbations are introduced to the underlying diffusion dynamics without affecting parity of the system, creating any long-range interactions, or bringing in additional conservation laws. Specifically, we study BARW and PCPD on a one dimensional periodic chain where the walkers hop to their nearest neighbor with a bias $\epsilon$. A walker at a given site diffuses towards its nearest neighbor with probability $\frac{1}{2}+\epsilon~{}(0\leq\epsilon\leq 1/2)$ and in the opposite direction with the complimentary probability $\frac{1}{2}-\epsilon$. For $\epsilon=0$, the walker performs a simple random walk and for $\epsilon=1/2$, the walker moves ballistically towards its nearest neighbor. It is to be noted that the bias on a walker at different sites are uncorrelated and so is the bias on the walker at different times. The process retains its Markovian nature and unlike the problem with Lev́y walker, there is no additional long-range interaction that is present in the system. However, the bias hinder the diffusion of walkers away from their parent cluster. Thus, for $\epsilon>0$, branching and annihilation within a cluster become the dominant processes. The case of annihilating random walkers ($i.e.$ no branching) in presence of the bias $\epsilon$ has been studied before sen_ray and it was found that the decay exponent changes from a value 1/2 without bias to 1 when bias is introduced. This suggests that under the bias, the random walkers at large times behave as ballistic walkers. When branching is turned on, there is an absorbing phase transition for $\epsilon=0$ pc_class ; pcpd_class . With $\epsilon>0$, one would expect that the transition between absorbing and active phases to occur at higher branching rates because of the enhanced annihilation. An important question to ask is whether this bias will affect the critical behavior of the transition and what are the possible universality classes it can give rise to. We study the problem using Monte Carlo simulations. We find that non-zero $\epsilon$ retains the universality class of DP, where as changes the PCPD class to DP and the PC class changes to a new universality class.

II Models

On a one dimensional lattice, the BARW with nearest neighbor bias $\epsilon$ ($0\leq\epsilon\leq 1/2$) is defined in the following way: with probability $p$, a random walker $A$ diffuses and with the complimentary probability $1-p$ it branches to produce $m(m>0)$ off-springs: $A\to(m+1)A$ at its nearest neighboring sites. When a walker diffuses, it does so with a probability $\frac{1}{2}+\epsilon$ towards its nearest neighboring walker and with probability $\frac{1}{2}-\epsilon$ in the opposite direction. When two walkers meet at the same site, they annihilate $(2A\to\O)$. For large values of $p$, all walkers get annihilated and the system goes to an absorbed state. When $p$ is small, the branching rate being higher, the system has a finite density of walkers even at large times and hence the system remains active forever. Therefore, by varying $p$, one can observe absorbing phase transitions in such systems at a particular critical value of $p=p_{c}$. When $m$ is even, the number of walkers modulo 2 is conserved at all times. This symmetry is called the parity. For $\epsilon=0$, the critical behavior of the transition between active and absorbing states depends upon parity. The critical behavior in these systems belong to PC when $m$ is even and to DP when $m$ is odd pc_class . We vary $\epsilon$ and find out how the critical behavior of the absorbing phase transitions change for $m=1$ and $m=4$ cases.

We also study the effect of the nearest neighbor bias on a binary process like PCPD pcpd_class , where branching can occur only when two random walkers are placed side by side. A walker is selected at random and its neigboring site (left or right) is chosen with equal probability. The system then evolves by following one of the three processes with the respective assigned probabilities as described below:

(i) with probability $q(1-D)$, the walker and its neighbor in the chosen site are annihilated $(2A\to\O)$.

(ii) if the chosen neighboring site is occupied and the next nearest site in the direction of the neighboring site is empty, a new walker is created ($2A\to 3A$) at the next nearest site with probability $(1-q)(1-D)$.

(iii) a walker diffuses with probability $D$ to one of its neighboring site (left or right) if it is empty. In this step the neighboring site is not chosen with equal probability. The target site for the diffusing walker is chosen with probability $1/2+\epsilon$ towards its nearest neigboring walker and with probability $1/2-\epsilon$ in the opposite direction.

For $\epsilon=0$, the critical point and the critical exponents for absorbing phase transition in PCPD generally depends upon both $q$ and $D$ pcpd_class . In this work, we find out the critical point $q_{c}$ for a fixed value of $D=1/2$ and study how the critical behavior changes as $\epsilon$ is varied.

III Simulation

To simulate BARW and PCPD, we start with a fully occupied lattice at time $t=0$ and measure the average density of walkers $\rho(t)=\langle s_{i}(t)\rangle$ as a function of time. Here, the $\langle\cdots\rangle$ represents average over configurations. The variable $s_{i}(t)$ takes the value $1$ when site $i$ is occupied by a walker and [math] when it is unoccupied. As $t\to\infty$, $\rho(t)$ saturates to a positive value $\rho_{a}$, if the system is in the active phase ($p<p_{c}$ or $q<q_{c}$) and decays to zero in the absorbing phase ($p>p_{c}$ or $q>q_{c}$). Thus, effectively, $\rho(t)$ acts as an order parameter for the system. At the critical point where $p=p_{c}$ for BARW and $q=q_{c}$ for PCPD, density decays with time as power law,

[TABLE]

$\alpha$ being the decay exponent. The critical points $p_{c}$, $q_{c}$ and the decay exponent $\alpha$ can be estimated by plotting $\rho(t)$ vs $t$ for different values of $p$ and $q$ respectively. At the critical point, a power law is obtained. For illustration, in Figures 1 and 2, we make an estimate of the critical point $q_{c}$ and the exponent $\alpha$ in the PCPD model for $\epsilon=0$ and $\epsilon=0.5$ respectively. For different values of $\epsilon$, the estimate of the critical point and $\alpha$ for BARW and PCPD has been compiled in Table 1.

The dynamical exponent $z$ can be determined from the finite size scaling analysis. In a finite system of size $L$, the decay of $\rho(t)$ as a function of $t$ at the critical point has the scaling form

[TABLE]

where $f$ is a scaling function. $f(x)$ is a constant for $x<1$ and decays exponentially for $x>1$. Once $\alpha$ has been measured, one can then determine $z$ using Eq. (2). At the critical point, the curves $\rho(t)t^{\alpha}$ vs $t/L^{z}$ for different values of $L$ collapse to a single curve. In Figures 3 and 4, we use the finite size scaling method to measure $z$ for PCPD when $\epsilon=0$ and $\epsilon$=0.5 respectively. The data for different values of $L$ collapses when $z=1.72$ for $\epsilon=0$ and $z=1.59$ for $\epsilon=0.5$

The order parameter exponent $\beta$ characterises the algebraic decay of the density $\rho_{a}$ as one approaches the critical point $q_{c}$ in the active phase $(q\to q_{c}^{-})$:

[TABLE]

Figure 5 shows the logarithmic plot of $\rho_{a}$ vs $q_{c}-q$ and estimated values of $\beta$ in PCPD for $\epsilon=0$ and $\epsilon=0.5$.

We also measure the two-point spatial correlations, $C(r)=\langle s_{i}(t)s_{i+r}(t)\rangle$, where $\langle\cdots\rangle$ is the configuration average. At the critical point, $C(r)$ decays as a power law,

[TABLE]

where the exponent $\theta=z\alpha$. In Figure 6, we plot $C(r)$ vs. $r$ in PCPD for $\epsilon=0$ and $\epsilon=0.5$, measured at $t=10^{5}$. In Table 1, we put the values of $\theta$ obtained from simulations for various values of $\epsilon$. The exponents $\theta$, $\alpha$ and $z$ seems to satisfy the scaling relation $\theta=z\alpha$ for all $\epsilon$.

IV Results and discussion

In BARW, for $m=1$ and $\epsilon=0$, our results match well with the DP exponents obtained previously pc_class . For any $\epsilon>0$, the exponents remain same as that for $\epsilon=0$. Therefore, we conclude that the bias does not affect the DP universality class. The BARW model for $m=4$ belong to the PC class pc_class in the absence of any bias $(\epsilon=0)$. Our results indicate the introduction of the bias drastically changes the exponents of the PC universality class. This is surprising because the bias does not the affect the parity of the system, neither creates any long-range interaction, nor does it gives rise to additional conservation laws. It would be interesting to know the universality class that the bias gives rise to for the PC class.

Our simulations show that the PCPD class crosses over to DP for any nonzero $\epsilon$. A characteristic feature of PCPD is the survival of solitary diffusing walkers for large times. When diffusion of single walkers is blocked in PCPD, its critical behavior is same as DP pcp_jensen . The bias $\epsilon$ tends to supress the diffusion of single walkers away from its parent cluster. For $\epsilon=1/2$, a single walker cannot leave a cluster and diffuse as a solitary walker. In Figure 7 we show the space-time plots for $\epsilon=0$ and $1/2$ in PCPD at the respective critical points. Clearly, the solitary diffusing walkers do not survive for large times for $\epsilon=1/2$ as compared to when $\epsilon=0$. In fact, this effect of solitary diffusing walkers not surviving for large times seems to happen for smaller $\epsilon$ values also. This is possibly the reason that for any $\epsilon>0$, PCPD crosses over to DP.

We conclude with the following observations. The universality class of DP is robust and remains unaffected by perturbations which alters the diffusion dynamics as long as the parity symmetry is unaltered. The same cannot be said for the PC and the PCPD universality classes. Parity alone does not guarantee the PC critical behavior, and the PC class may change under pertubations of the diffusion dynamics. But, as long as parity is kept unchanged, PC class does not seem to go to DP class. The PCPD critical behavior is rather unstable. Although it is not affected by parity, perturbations which arrests the diffusion of single walkers makes it cross over to DP.