Tagged particle dynamics in one dimensional A+A→kA models with the particles biased to diffuse towards their nearest neighbour
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Tagged particle dynamics in one dimensional A + A → kA models with the particlesbiased to diffuse towards their nearest neighbour Reshmi Roy, Purusattam Ray,
2, 3 and Parongama Sen Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India. Institute for Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India. Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India.
Dynamical features of tagged particles are studied in a one dimensional A + A → kA system for k = 0 and 1, where the particles A have a bias ǫ (0 ≤ ǫ ≤ .
5) to hop one step in the direction oftheir nearest neighboring particle. ǫ = 0 represents purely diffusive motion and ǫ = 0 . ǫ , there is a time scale t ∗ whichdemarcates the dynamics of the particles. Below t ∗ , the dynamics are governed by the annihilationof the particles, and the particle motions are highly correlated, while for t ≫ t ∗ , the particles moveas independent biased walkers. t ∗ diverges as ( ǫ c − ǫ ) − γ , where γ = 1 and ǫ c = 0 . ǫ c is a criticalpoint of the dynamics. At ǫ c , the probability S ( t ), that a walker changes direction of its path attime t , decays as S ( t ) ∼ t − and the distribution D ( τ ) of the time interval τ between consecutivechanges in the direction of a typical walker decays with a power law as D ( τ ) ∼ τ − . I. INTRODUCTION
Reaction diffusion systems in their simplest form withdiffusion and annihilation of particles have been studiedover the years [1–4]. These are nonequilibrium systemsof diffusing particles undergoing certain reactions. De-pending on the nature of the problem, the particles couldbe molecules, chemical or biological entities, opinions insocieties or market commodities. Such systems are fre-quently used to describe various aspects of wide varietiesof chemical, biological and physical problems. In the lat-tice version of the single species problem, the lattice isfilled with particles (say A ) with some probability ini-tially and at each time step, the particles are allowedto jump to one of the nearest neighbouring sites (diffu-sion) with a certain probability. The simplest form ofparticle reaction is when a certain number l of the par-ticles meet: lA → kA with k < l . It is well known thatannihilating random walkers with l = 2 and k = 0 corre-sponds to the Ising-Glauber kinetics while the coalescingcase with l = 2 and k = 1 describes the dynamics ofthe q state Potts model with q → ∞ , both at zero tem-perature and in one dimension [5]. Such systems havebeen studied in one dimension [6–13] as well as in higherdimensions [14–17]. Depending on the initial condition,whether one starts with even or odd number of particles,the steady state will contain no particles or one particlerespectively. The focus in all these analysis is how thesystem approaches the steady state. In particular, oneintends to know how the number of particles decays withtime and the distribution of the intervals between theparticles evolves with time.Various reaction diffusion systems have been studiedwith different values of k and l in the past for different dy-namical processes like ballistic annihilation[18–20], Levywalks [21, 22] and of course simple diffusion. However,what happens if the dynamical process is intrinsicallystochastic and diffusive is an important question whichhas not been studied much. The idea behind all these studies is to find any universal behaviour in these sys-tem and the key factors which determine the universal-ity. Here we ask the same question by introducing a biaswhich does not alter the existing features like conserva-tion, range and nature of the interaction or the diffusionaldynamics in the model.We have studied the model A + A → kA where theparticle A diffuses with a preference towards its nearestneighbour. Both the annihilating case ( k = 0) and thecoalescing case ( k = 1) have been considered. It is im-portant to note that this bias does not affect the annihi-lation process and retains the Markovian property of thedynamics. This simple extension, indeed, leads to drasticchanges in the bulk dynamical features. For k = 0, thefraction of walkers ρ ( t ) at time t was found to decay as ρ ( t ) ∼ t − α , where α ≈ α = 1 /
2. The value of α suggests that in thepresence of the bias, the walkers, in the long time limit,behave as ballistic walkers.For the coalescing case withbias, the bulk behaviour is identical, i.e., α ≈ k = 0 and 1), specifically to check whetherthey perform ballistic motion or not.In the following we briefly introduce the models andmention the different features studied and also the mainresults obtained. We have a bunch of walkers on a onedimensional ring. At every time step, the walker hopsone step to its left or right with a bias ǫ to move inthe direction of its nearest neighbour. ǫ = 0 implies nobias so that the walkers are purely random walkers and ǫ = ǫ c = 0 . k = 0, we have a moredetailed presentation of the results. First, the probability P ( x, t ) that a particle is at a distance x from its originafter time t is estimated. We then calculate the proba-bility that a walker changes its direction as a function oftime. The distribution of the time intervals over whichthe walk continues in the same direction is also obtained.A change in the direction of motion can occur either dueto diffusion or annihilation of the nearest particle(s). Wefind that the dynamics of the walkers are controlled bytwo time regimes. For time t < t ∗ , the dynamics are con-trolled by the annihilation of the particles. The motionof the walkers, in this regime, is highly correlated and theprocess is critical in the sense that there is no time scalein it. As a result, the probability S ( t ) of the change inthe direction of the motion of the walker at time t decayswith a power law; S ( t ) ∼ /t . Similarly, the distribution D ( τ ) of the time interval τ spent between two changesin the direction of the motion of the walkers is scale freeas D ( τ ) ∼ /τ . We have found the full scaling behav-ior and arguments for the values of the exponents. Thecrossover time t ∗ ∼ ( ǫ c − ǫ ) − , so ǫ c can be interpreted asa dynamical critical point where a diverging time scaleexists.We have also studied the coalescing model ( k = 1) withsimilar bias, i.e., A + A → A model. Without the bias, itis equivalent to the A + A → ∅ model as far as the decayof particles in time is concerned. In presence of bias,the scaling of the fraction of surviving particles ρ ( t ) ∝ t − (details in section IV) shows that it is similar to theannihilating model. The dynamics of the particles areindeed different in the coalescing model as the distancesbetween the particles are not much affected by a reaction,except for the surviving particle that remains after thereaction. Here we have focussed on the behaviour of S ( t )and D ( τ ) and find that the qualitative features of thedynamics of the tagged particle are again the same as inthe A + A → ∅ model. However, here the crossover tothe diffusion behaviour occurs at later times, so that t ∗ ishigher in the A + A → A model. This is consistent withour inference that the early time regime is annihilationdominated as for k = 1, the annihilation continues for alonger time. II. THE MODEL, DYNAMICS ANDSIMULATION DETAILS
The model consists of walkers denoted by A , under-going the reaction A + A → kA . At each update, asite is selected randomly and if there is a particle on it,it moves towards its nearest neighbour with probability ǫ + 0 . ≤ ǫ ≤ .
5) and otherwise in the opposite di-rection. For k = 0, if there is already another walkerlocated on this neighbouring site, then both particles areannihilated and for k = 1, one of them survives. Sup-pose, a walker is at site i and its nearest neighbours are at i + x and i − y on its right and left respectively; thewalker will hop one step towards right with probability0 . ǫ and to left with probability 0 . − ǫ if x < y . In therare cases where the two neighbours are equidistant, thewalker moves in either direction with equal probability.When the bias ǫ = 0 .
5, the k = 0 case correspondsto the spin model introduced in [27] (see Appendix fordetails). Hence, the dynamical updating scheme usedhere for k = 0 has a one to one correspondence with theoriginal spin dynamics used in [27]. As the spin systemin [27] was considered to be highly disordered initially,we start with a high density of walkers in this problem;specifically the number of walkers is chosen to be L/ L . To maintain thecorrespondence with spin dynamics, the walkers are up-dated asynchronously and at each update a site is cho-sen randomly, rather than a walker, for updating. OneMonte Carlo step (MCS) comprises of L such updates.The same dynamical scheme was used in [23, 24] wherethe bulk properties of the walker model were studied.The dynamical scheme allows the possibility that awalker’s state may not be updated at all. This is be-cause if a site is not selected, the position of the corre-sponding walker will not be updated. It may also happenthat a walker is updated more than once in the followingmanner: if a walker moves to the site j and the site j is selected later, then the position of the same walker isupdated again. This signifies that the net displacementof a particular walker may even be zero when it per-forms more than one movement in the same MCS. Forall calculations, the final positions of the walkers afterthe completion of one MCS are considered. The resultsreported here are for simulations done on lattices of size12000 or more and the maximum number of configura-tion studied was 2000. Periodic boundary condition hasbeen used for all the simulations.In the A + A → A model ( k = 1), the same dynam-ical scheme is used. Here, once two walkers meet, oneof them will survive. In order to study the tagged par-ticle dynamics, we need to label the surviving particle.We use the convention that the particle which makes thelast movement survives. We have checked that the ran-dom convention (either of the two particles is taken to bethe survivor randomly) leads to the same results qualita-tively. III. RESULTS FOR A + A → ∅ ( k = 0) MODEL
To check the movement of individual walkers we tooksnapshots of the system at different times. Fig. 1(a) and(b) show the world lines of the motion of the particles for ǫ = 0 and ǫ = 0 .
5. It clearly shows that the motion of theindividual particles in the two extreme cases are remark-ably different. Annihilation dominates for ǫ = 0 . ǫ = 0 the walk is diffusive as expected. For the in-termediate values of ǫ , both the mechanisms of diffusionand annihilation will be important and thus, as we will
40 80 120 160 200 0 40 80 120 160 200 (a) t i m e (b) (c) t i m e space (d) FIG. 1. Snapshots of the system at different times for ǫ =0(a) and ǫ =0.5 (b) for A + A → ∅ model. Lower panel showthe snapshots for A + A → A model for ǫ =0 (c) and ǫ =0.5(d). The trajectories in different colors represent differentparticles. see later, give rise to the crossover effect for the system.To probe the dynamics of the particles, we have studiedthe following three quantities: (i) the probability distri-bution P ( x, t ) of finding a particle A at distance x fromits origin at time t , (ii) the probability S ( t ) of the changein the direction in the motion of a walker at a time t and(iii) the distribution D ( τ ) of the time interval τ betweentwo successive changes in the direction of the motion ofa walker. The results for each of these quantities aredescribed in the following three subsections. A. Probability distribution P ( x, t ) For ǫ = 0, the single particle motion is diffusive and thecorresponding probability distribution P ( x, t ) is knownto be Gaussian. This remains true even in the presenceof annihilation.For ǫ = 0, P ( x, t ) changes drastically. The distribu-tions are still symmetric as the motion of individual par-ticles can occur in both directions (left and right). How-ever, there is no peak at the origin ( x = 0) and insteada double peak structure emerges with a dip at x = 0. Toobtain a collapse of the data at different times, we notethat the scaling variable is x/ǫt for all values of ǫ . Wefind that the collapsed data can be fit to the form P ( x, t ) ǫt = f ( xǫt ) ∝ exp[ − β { ( γxǫt ) − } ] . (1)The data collapse in the early time regime is shown inFig. 2a. However, the data collapse as well as the abovescaling form seems to be less accurate at later times. On P ( x ,t ) ε t x/( ε t) ε =0.2,t=100 ε =0.2,t=200 ε =0.3,t=50 ε =0.3,t=150 ε =0.3,t=200f(x) P ( x ,t ) ε t x/( ε t) ε =0.1,t=1000 ε =0.1,t=2000 ε =0.2,t=500 ε =0.2,t=1500 ε =0.5,t=600 ε =0.5,t=1200f(x) FIG. 2. (a): Data collapse of P ( x, t ) ǫt against x/ǫt for ǫ = 0 . . P ( x , t ) ǫ t against x /ǫ t are shown for ǫ = 0 . , . , . A + A → ∅ model. investigating further, we find that while we attempt tofit the data individually for each ǫ and t by the formgiven in Eq. (1), only in the early regime ( ǫt . β and γ are constants. While γ shows negligibledependence on ǫ and t , β strongly depends on ǫt ; beyond ǫt = 100 it is no longer a constant but increases sharplyas a function of ǫt . Hence the distribution scaled in theabove manner shows a dip at the center which goes downwith time while the peak heights increase such that thedata do not collapse well as shown in Fig. 2b.The above study suggests that at vary late stages, thescaled distribution will assume a double delta functionalform and a universal scaling function exists only in theearly time regime ( ǫt . B. Probability of change in direction
The probability of direction change at time t is ob-tained by estimating the fraction of walkers that changedirection at time t. For ǫ = 0, as the system is diffusive,the probability of direction change S ( t ) = p , a constantindependent of time. For a purely diffusive random walk, p = 0 .
5. But here asynchronous dynamics have beenused and this updating scheme allows the walkers to re-main in the same state within a MCS as already discussedin the previous section. This dynamics can only decreasethe probability of change in direction. p for ǫ = 0 actu-ally turns out to be ≈ .
27 numerically.For 0 < ǫ < .
5, the change in direction of a walkeroccurs due to two reasons; either due to the annihilationof a neighbouring walker or because of the diffusive com-ponent which is large for small ǫ . At earlier times, thewalker density is large and so the number of annihilationis considerable. Therefore the change in direction of thewalkers is dominated by the annihilation process. How-ever, as time progresses, annihilation becomes rarer andtherefore the diffusive component becomes the dominat- -5 -4 -3 -2 -1
1 10 100 1000 S ( t ) t ε = 0ε = 0.2ε = 0.4ε = 0.45ε = 0.47ε = 0.49ε = 0.5 f(t)=0.4t -1.07 -2 -1 S s a t ε c - ε S sat FIG. 3. Probability of direction change of tagged particle fordifferent ǫ in the A + A → ∅ model. For ǫ = 0 .
5, it decays as t − . Inset shows the variation of S sat with ǫ c − ǫ . ing factor. So a saturation value S sat of S ( t ) is reachedat a later time, typically after a time t ∗ . The data for S ( t ) is shown in Fig. 3 and the inset shows the variationof S sat with ǫ c − ǫ where ǫ c = 0 .
5. As expected, S sat decreases as ǫ is made larger. In fact, we find that unless ǫ is very close ǫ c , the saturation is reached very fast, typ-ically within one hundred MC step. S sat shows a linearvariation with ǫ c − ǫ , shown in inset of Fig. 3.One can obtain a data collapse by plotting S ( t ) t against t ( ǫ c − ǫ ), shown in Fig. 4. This indicates thatone can write S ( t ) as S ( t ) = 1 t g ( z ) , (2)where z = t ( ǫ c − ǫ ) and g ( z ) is a scaling function. g ( z ) isconstant for z < g ( z ) ∼ z for large z . Therefore, p ǫ ≡ S ( t → ∞ ) ∝ ( ǫ c − ǫ ), which is consistent with thevariation of S sat with ( ǫ c − ǫ ) (see inset of Fig. 3). Henceone can argue that t ∗ = ( ǫ c − ǫ ) − acts as a timescale,below which S ( t ) ∝ t − δ with δ = 1. As t ∗ diverges at ǫ c ,there is no saturation region for ǫ = ǫ c and S ( t ) showsa power law decay, S ( t ) ∼ t − for all times as shown inFig. 3. The divergence of t ∗ as ǫ → ǫ c justifies that ǫ c isthe dynamical critical point.One can argue that the value of δ is unity for the de-terministic case ǫ = 0 .
5, where the walker always movestowards its nearer neighbour. A direction change canoccur only if an adjacent walker is annihilated (how-ever, this is a necessary but not sufficient condition).Let A ( t ) be the number of annihilation taking place attime t . If N ( t ) is the number of walkers at time t , A ( t )is given by − dNdt ∝ t − α − = t − . Since N ( t ) is pro-portional to t − and S ( t ) is proportional to A ( t ) /N ( t ),therefore S ( t ) ∼ t − . It may be added here that A ( t )and N ( t ) have the same behaviour for all ǫ = 0, how-ever, for ǫ = 0 .
5, direction change may occur even whenthere is no annihilation. The above argument is validonly for ǫ = 0 . -2 S ( t ) t t| ε - ε c | ε = 0.46, A+A → ϕε = 0.47, A+A → ϕε = 0.48, A+A → ϕε = 0.49,A+A → ϕ ε = 0.46, A+A → A ε = 0.47, A+A → A ε = 0.48, A+A → A ε = 0.49, A+A → A0.085x
FIG. 4. Variation of S ( t ) t with t | ǫ − ǫ c | shows a data collapsefor both the models A + A → ∅ and A + A → A , where ǫ c = 0 .
5. Data for the A + A → A model have been shiftedalong y axis for better clarity. The linear regions in the log-log plot are fitted to power law forms with the exponent veryclose to unity. nent. However, the fact that S ( t ) ∝ t − in the earlytime regime for ǫ = 0 . t ∗ , the bias is not strong enough to causetwo particles to come close enough and cause an anni-hilation. The motion effectively becomes uncorrelated.Obviously, the crossover occurs at later times as ǫ , rep-resenting the bias, becomes larger and the inherent dif-fusive component becomes weaker making annihilationsmore probable. Therefore, at ǫ = 0 .
5, the fully biasedpoint, S ( t ) ∝ t − and the crossover time diverges.The nature of the walk remains ballistic in all regimesdue to the bias, however small, to move towards the near-est neighbours. This is consistent with the conjecturethat the probability distribution assumes a double deltaform at large times mentioned in the last subsection. C. Distribution of time intervals betweenconsecutive change in direction
Another interesting quantity is D ( τ ), the interval oftime τ spent without change in direction of motion. Forrandom walkers with ǫ = 0, the probability that in thetime interval τ , there is no direction change is given by D ( τ ) = p (1 − p ) τ . (3)This reduces to an exponential form: D ( τ ) ∝ exp[ − τ ln { / (1 − p ) } ]. Fig. 5a shows the data for D ( τ ) for ǫ = 0. From the numerical simulation, we find D ( τ ) ∼ exp( − τ ln 1 .
38) for ǫ = 0, which is consistentwith p ≈ . ǫ = 0, we note that D ( τ ) obeysthe following form D ( τ ) ∼ τ φ ( z ) , (4)where z = τ ( ǫ c − ǫ ) is the scaling argument and φ ( z )is the scaling function. φ ( z ) is constant for z < − z ) for z ≫
1. The data are shownin Fig. 5b.Thus it is indicated that here also a crossover be-haviour occurs at τ = τ ∗ with τ ∗ ∝ ( ǫ c − ǫ ) − , be-yond which the exponential decay is observed and belowwhich there will be a power law behaviour. Obviously for ǫ = 0 . τ ∗ diverges such that only the power law decaywill be observed with an exponent 2 which is indeed thecase as shown in Fig. 5c.It can be argued why the exponent is 2 for ǫ = 0 . t + 1 to t + τ . This means it changes direc-tion at times t and t + τ + 1. Hence, D ( t , τ ) is givenby D ( t , τ ) = S ( t ) S ( t + τ + 1) τ Y x =1 [1 − S ( t + x )] . Using the variation of S ( t ) ∝ /t obtained in the lastsubsection, D ( t , τ ) ∝ ( t − )( t + τ + 1) − τ Y x =1 (1 − t + x ) . (5)Taking logarithm of both sides of Eq. (5) and convert-ing summation into an integral, one getsln D ( t , τ ) = − t + τ ) , apart from a constant factor. One can always choose theorigin t to be zero, such that D ( τ ) ∼ τ − (6)showing consistency with the numerical results. (Fig.5b).One can also justify the crossover behaviour for 0 <ǫ < .
5. Here, the crossover behaviour in S ( t ) found inSec. III B, should be taken into account while calculat-ing D ( τ ). S ( t ) decays in a power law manner at shorttimes to a constant value in the late time regime. Therelatively larger value of S ( t ) will be responsible for thebehaviour of D ( τ ) for small τ . Hence, for small τ , thepower law behaviour of S ( t ) will be relevant for which ithas been already shown that D ( τ ) ∝ τ − . On the other hand, the constant (lower) value of S ( t ) will be respon-sible for contribution to D ( τ ) for large values of τ . For t > t ∗ , S ( t ) = p ǫ = a ( ǫ c − ǫ ), where a is a constant lessthan unity (see Fig. 4). Using this value, one gets there-fore D ( τ ) = ( p ǫ ) (1 − p ǫ ) τ ∼ exp( τ ln(1 − a ( ǫ c − ǫ ))).As a ( ǫ c − ǫ ) is less than unity, the expression for D ( τ )simplifies to D ( τ ) ∼ exp( − a ( ǫ c − ǫ ) τ ) . (7) D ( τ ) can indeed be fit to an exponential form for largevalues of τ (see Fig. 5d): D ( τ ) ∼ exp( − bτ ) (as long as ǫ is not very close to ǫ c for reasons that will be clarifiedlater) and b can be fitted to the form b = b ( ǫ c − ǫ ) , (8)where b = 0 .
5, shown in Fig. 6. This agrees with theexpectation that b should be varying linearly with ( ǫ c − ǫ ) as indicated by Eq. (7). It is also observed that b approaches the value ln 1 .
38 as ǫ → b → ǫ → ǫ c = 0 . ǫ approaches 0.5 the crossovertime increases and the exponential behaviour exists onlyfor very large values of τ where the statistics is obviouslypoorer. This is the reason for which the estimation of b for ǫ → ǫ c becomes less reliable as mentioned before. Onthe other hand, to show the power law region one has touse values of ǫ fairly close to 0.5. IV. RESULTS FOR A + A → A ( k = 1 ) MODEL For the A + A → A model, the ǫ = 0 case is known tohave the scaling form for the fraction of surviving par-ticles as ρ ( t ) ∝ t − / [18]. In the biased case, with any ǫ = 0 we find that the scaling is again like the A + A → ρ ( t ) ∝ t − shown in Fig. 7. Typicalsnapshots of the walk are shown in Fig. 1(c) and (d).For the motion of the tagged particles in the A + A → A model, we restrict the study to the probability of direc-tion change and distribution of the time interval of mo-tion executed without direction change. Again we find nosignificant change from the behaviour for the A + A → S ( t ) ∝ t − for ǫ = 0 . ǫ , there is a crossover to a diffusive behaviour.In fact, when S ( t ) t is plotted against t ( ǫ c − ǫ ), we againfind that the scaling function has a constant part anda linear variation at larger values of the scaled variable(Fig. 4).One can, in fact, use the same argument to justify thescaling behaviour S ( t ) ∝ t − for ǫ = 0 .
5. This is becausein this case also, the only way the direction change cantake place is through annihilation. However, there is asubtle difference. For the A + A → ∅ model, when twoparticles are annihilated, direction change can take placefor their neighbouring particles. On the other hand, in -6 -4 -2
0 10 20 30 40 50 (a) D ( τ ) τ ε =0h(x)=0.38exp(-xln 1.38) -1 (b) τ D ( τ ) τ ( ε c - ε ) -5 -3 -1
10 30 100 300 1000 (c) D ( τ ) τ ε =0.41 ε =0.45 ε =0.5x -2 -7 -5 -3 -1
10 30 50 70 (d) D ( τ ) τ ε =0.2f(x)=0.54exp(-0.17x) FIG. 5. (a), (c) and (d) show data for the A + A → ∅ modelwhile (b) includes the data for the A + A → A model also.(a) Variation of D ( τ ) over τ for ǫ = 0 (b) Collapsed datafor D ( τ ) τ against τ ( ǫ c − ǫ ) for ǫ = 0 . , . , . A + A → A model (lower curves) have been shiftedfor better clarity. (c) shows the data for D ( τ ) against τ for ǫ = 0 . , . , .
5, where power law decay exists over a smalltime interval and power law region decreases with the decreaseof ǫ . (d) shows the variation of D ( τ ) against τ for ǫ = 0 . τ ≫ t ∗ for A + A → ∅ model. ε b , b ′ ε c - ε bb ′ ε c - ε ) b , b ′ bb ′ FIG. 6. Variation of b and b ′ with ǫ c − ǫ shown in a log-logplot when ǫ is very close to 0.5 and inset shows the variationwith ǫ for the full range. the A + A → A case, the direction change may occurfor the surviving particle while its neighbouring parti-cles usually remain unaffected (see Fig. 1). Anotherimportant point to note is that in the scaling functionfor S ( t ) t , the linear fitting is appropriate beyond a largervalue of the scaled variable, i.e., the crossover to diffu-sive behaviour takes place later in the A + A → A modelin comparison (see Fig. 4). This is consistent with ourinference that the early time regime is annihilation dom- -4 -3 -2 -1
1 10 100 1000 ρ ( t ) t ε =0, A+A → ϕε =0.5, A+A → ϕε =0, A+A → A ε =0.3, A+A → A ε =0.5, A+A → Ax - 0.5 x -1 FIG. 7. Variation of ρ ( t ) with t is shown in a log-log plot fordifferent values of ǫ for A + A → ∅ and A + A → A models. inated as the annihilations in the A + A → A continuefor a longer time.The distribution for the time intervals of motion with-out change in direction again shows similar scaling. InFig. 5(b), we show the comparative behaviour for thetwo models. The tail of the scaling function is obtainedonce again as exp( − b ′ τ ), where b ′ shows a linear variationwith ( ǫ c − ǫ ) (Fig. 6). V. DISCUSSIONS
We have studied the motion of the tagged particles A ,in one dimension, undergoing the reaction A + A → kA with k = 0 and 1 with the additional feature that a par-ticle walks with a probability 0 . ǫ towards its nearestneighbour and with a probability 0 . − ǫ in the other di-rection. This is perhaps one of the simplest models whichexhibits critical dynamics.The particles, when ǫ = 0, perform normal randomwalk, so their motions are not correlated. The reactionmakes the fraction ρ of particles decay with time t as N ( t ) ∼ t − α with α = 1 /
2. For any non-zero ǫ , the valueof α has been found to be altered to 1. The value of α = 1 suggests that the particle motion is not randomanymore but is ballistic. However, it has to be remem-bered that A + A → ∅ model with ballistic walkers A donot correspond to α = 1 and the results depend on thedistribution of initial velocities of the particles [18, 19].Studying the tagged particles reveal that the effect of ǫ in conjunction with the annihilation reaction makes thedynamics of the particles correlated over a large timescale. This time scale depends on ǫ and diverges at ǫ =0 .
5. Consequently, the dynamics become critical, in thesense that, the probability S ( t ) of the particles to changethe direction of their motions reduces with time as 1 /t and the distribution D ( τ ) of time interval τ over whichthe particles on average move along the same directionfollows power law: D ( τ ) ∼ /τ .Detailed study of S ( t ) and D ( τ ) shows that there isa crossover from the annihilation dominated regime toa (partially) diffusive regime at time t ∗ ∝ ( ǫ c − ǫ ) − .Beyond t ∗ , S ( t ) is a constant for 0 < ǫ < .
5, althoughthe actual value is less compared to the unbiased case ǫ =0. However, the overall motion is still ballistic, h| x |i ∼ t ,for any ǫ > P ( x, t )tending towards a double delta function (studied for the k = 0 model) at very late times while for ǫ = 0, thedistribution is always Gaussian.It may be mentioned here that the change in the be-haviour of the probability distribution from a Gaussianfor ǫ = 0 to a bimodal form for ǫ = 0 is reminiscent of theorder parameter distribution above and below the criticaltemperature for Ising like systems; the form in Eq. (1) isalso similar to the case for continuous spins.In conclusion, we have shown how the bias to movetowards nearest neighbours generates correlation in themotion of the particles in a simple A + A → kA reaction process. Also, we conclude that the divergences in thetimescales and power law behaviour in the relevant dy-namical variables indicate that ǫ c = 0 . ǫ = 0 .
5, the motion is still stochastic. The presentstudy is able to manifest at the individual level the pre-cise role of the bias and how the dynamics are differentfrom simple ballistic motion.Acknowledgement: The authors thank DST-SERBproject, File no. EMR/2016/005429 (Government ofIndia) for financial support. Discussions with SohamBiswas is also acknowledged. [1] Privman V., ed.
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