Spatial structure in diffusion-limited two-species annihilation

Logarithmic speed-up of relaxation in A-B annihilation with exclusion

We show that the decay of the density of active particles in the reaction A+B→0 in one dimension, with exclusion interaction, results in logarithmic corrections to the expected power law decay, when the starting initial condition (i.c.) is periodic. It is well known that the late-time density of surviving particles goes as t^{-1/4} with random initial conditions, and as t^{-1/2} with alternating initial conditions (ABABAB⋯). We show that the decay for periodic i.c.'s made of longer blocks (A^{n}B^{n}A^{n}B^{n}⋯) do not show a pure power-law decay when n is even. By means of first-passage Monte Carlo simulations, and a mapping to a q-state coarsening model which can be solved in the independent interval approximation (IIA), we show that the late-time decay of the density of surviving particles goes as t^{-1/2}[ln(t)]^{-1} for n even, but as t^{-1/2} when n is odd. We relate this kinetic symmetry breaking in the Glauber Ising model. We also see a very slow crossover from a t^{-1/2}[ln(t)]^{-1} regime to eventual t^{-1/2} behavior for i.c.'s made of mixtures of odd- and even-length blocks.

Extinction and survival in two-species annihilation

We study diffusion-controlled two-species annihilation with a finite number of particles. In this stochastic process, particles move diffusively, and when two particles of opposite type come into contact, the two annihilate. We focus on the behavior in three spatial dimensions and for initial conditions where particles are confined to a compact domain. Generally, one species outnumbers the other, and we find that the difference between the number of majority and minority species, which is a conserved quantity, controls the behavior. When the number difference exceeds a critical value, the minority becomes extinct and a finite number of majority particles survive, while below this critical difference, a finite number of particles of both species survive. The critical difference Δ_{c} grows algebraically with the total initial number of particles N, and when N≫1, the critical difference scales as Δ_{c}∼N^{1/3}. Furthermore, when the initial concentrations of the two species are equal, the average number of surviving majority and minority particles, M_{+} and M_{-}, exhibit two distinct scaling behaviors, M_{+}∼N^{1/2} and M_{-}∼N^{1/6}. In contrast, when the initial populations are equal, these two quantities are comparable M_{+}∼M_{-}∼N^{1/3}.

Absence of absorbing phase transitions in a conserved lattice-gas model in one dimension

A one-dimensional conserved lattice-gas model is known to undergo continuous absorbing phase transitions where some of the critical exponents are exactly known. In one dimension, we recently showed that the model is mapped onto a two species reaction A+B→0 with diffusion rate of D_{A}>0 and D_{B}=0. In this work, it is explicitly shown from the scaling theory for A+B→0 that the observed scaling behavior of the conserved lattice-gas model is not associated with the absorbing phase transitions. Instead, the model indeed undergoes a crossover between two different scaling behaviors of A+B→0, the scaling behaviors of equal and unequal initial densities of two species. The crossover is similar to the absorbing transitions in many respects but some important features of continuous transitions such as the diverging fluctuations of an order parameter are absent.

Effects of random initial conditions on the dynamical scaling behaviors of a fixed-energy Manna sandpile model in one dimension

For a fixed-energy (FE) Manna sandpile model in one dimension, we investigate the effects of random initial conditions on the dynamical scaling behavior of an order parameter. In the FE Manna model, the density ρ of total particles is conserved, and an absorbing phase transition occurs at ρ(c) as ρ varies. In this work, we show that, for a given ρ, random initial distributions of particles lead to the domain structure in which domains with particle densities higher and lower than ρ(c) alternate with each other. In the domain structure, the dominant length scale is the average domain length, which increases via the coalescence of adjacent domains. At ρ(c), the domain structure slows down the decay of an order parameter and also causes anomalous finite-size effects, i.e., power-law decay followed by an exponential one before the quasisteady state. As a result, the interplay of particle conservation and random initial conditions causes the domain structure, which is the origin of the anomalous dynamical scaling behaviors for random initial conditions.

Block renormalization study on the nonequilibrium chiral Ising model

We present a numerical study on the ordering dynamics of a one-dimensional nonequilibrium Ising spin system with chirality. This system is characterized by a direction-dependent spin update rule. Pairs of +- spins can flip to ++ or -- with probability (1-u) or to -+ with probability u while -+ pairs are frozen. The system was found to evolve into the ferromagnetic ordered state at any u<1 exhibiting the power-law scaling of the characteristic length scale ξ∼t(1/z) and the domain-wall density ρ∼t(-δ). The scaling exponents z and δ were found to vary continuously with the parameter u. To establish the anomalous power-law scaling firmly, we perform the block renormalization analysis proposed by Basu and Hinrichsen [U. Basu and H. Hinrichsen, J. Stat. Mech.: Theor. Exp. (2011)]. The block renormalization method predicts, under the assumption of dynamic scale invariance, a scaling relation that can be used to estimate the scaling exponent numerically. We find the condition under which the scaling relation is justified. We then apply the method to our model and obtain the critical exponent zδ at several values of u. The numerical result is in perfect agreement with that of the previous study. This study serves as additional evidence for the claim that the nonequilibrium chiral Ising model displays power-law scaling behavior with continuously varying exponents.

Comment on "Dependence of asymptotic decay exponents on initial condition and the resulting scaling violation"

We present the mapping relation between a one-dimensional conserved lattice gas model and a two species reaction A+B→0. From the kinetics of A+B→0, we show that the anomalous critical decay of an order parameter in the conserved lattice gas model results from finite-size effects induced by the domain structure for random initial conditions where the scaling relation z=ν∥/ν⊥ is violated. A simple initial condition satisfying the scaling relation is also discussed.

Critical behavior of absorbing phase transitions for models in the Manna class with natural initial states

The critical behavior of absorbing phase transitions for two typical models in the Manna universality class, the conserved Manna model and the conserved lattice gas model, both on a square lattice, was investigated using the natural initial states. Various critical exponents were estimated using the static and dynamic simulations. The exponents characterizing dynamics of active particles differ considerably from the known exponents obtained using the random initial states, whereas those associated with the steady-state quantities remain the same. The critical exponents for both models were consistent with errors of less than 1% and satisfied the known scaling relations; thus, the known violation of scaling relations for models with a conserved field was resolved using the natural initial states. The results differed by 7%∼12% from the directed percolation values.

Reaction-diffusion processes with nonlinear diffusion

We study reaction-diffusion processes with concentration-dependent diffusivity. First, the decay of the concentration in the single-species and two-species diffusion-controlled annihilation processes is determined. We then consider two natural inhomogeneous realizations. The two-species annihilation process is investigated in the situation when the reactants are initially separated, namely each species occupies a half space. In particular, we establish the growth law of the width of the reaction zone. The single-species annihilation process is studied in the situation when the spatially localized source drives the system toward the nonequilibrium steady state. Finally, we investigate a dissolution process with a localized source of diffusing atoms which react with the initially present immobile atoms forming immobile molecules.

Irreversible nA+mB→0 reaction of driven hard-core particles

We investigate the kinetics of general two species annihilation nA+mB→0 of driven hard-core (HC) particles with N=n+m in one dimension. With uniform drift velocity, all particles are driven to the right. HC exclusion forbids the interchange of any particles and restricts the number of particles on a site to 0 or 1. The reaction is classified into two classes, the symmetric and the asymmetric reaction. The symmetric reaction means both nA+mB→0 and mA+nB→0 , while the asymmetric reaction means only nA+mB→0 for a given (n,m) pair. As N increases, the trains of particles causing the reaction rarely form. Hence, for sufficiently large N, particles are evenly distributed before the reaction, so one expects a crossover N(c) above which the kinetics follows the classical mean-field rate equation. We show the existence of N(c) and that the kinetics for N<N(c) is the same as that of A+B→0 of driven HC particles as in the reactions with the isotropic diffusion. However, compared to the isotropic cases, N(c) and the kinetics for N≥N(c) are shown to be completely changed by the interplay of the drift and HC exclusion, and strongly depend on the reaction symmetry. We also show that densities decay as t(-1/N) which cannot be explained by the classical mean-field rate equation. Instead the kinetics is explained analytically by a variant theory.

Reaction symmetry of irreversible reaction nA+mB-->0

We investigate the kinetics of irreversible reaction nA+mB-->0. When nA particles and mB particles encounter on the same site, the reaction takes place. We classify the reaction into two classes, the symmetric and the asymmetric reactions. The symmetric reaction means both nA+mB-->0 and mA+nB-->0, while the asymmetric reaction means only nA+mB-->0 for a given (n,m) pair. The kinetics of the reaction follows the fluctuation-dominated kinetics of A+B-->0 for N<Nc, where Nc is the crossover N(=n+m). For N>or=Nc, the kinetics follows a mean-field rate equation. For the asymmetric reaction, it was shown that Nc is 5. We numerically show that the reaction symmetry changes Nc in one dimension. We investigate the asymptotic scaling behaviors of density and various lengths characterizing the spatial organization of particles such as domain length and interparticle distance. Lengths exhibit much clear crossover to the mean-field region at the expected Nc of each reaction. From the scaling behavior of density and lengths, we show Nc=4 for the symmetric reaction in one dimension. Therefore the symmetry in the reaction changes the upper critical dimension, which implies that the kinetics cannot be described by a single theory.

Fast chemical reaction and multiple-scale concentration fields in singular vortices

Two species involved in a simple, fast reaction tend to become segregated in patches composed of a single of these reactants. These patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs, fed by diffusion. Stirred by advection, this boundary and the concentration fields within the patches may tend to present multiple-scale characteristics. Based on this segregated state, this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it, when concentrations presenting initial self-similar fluctuations are advected by a singular vortex. First the two sources of singularity, i.e., the self-similar initial conditions and the singular vortex, are considered separately. On the one hand, self-similar initial conditions are imposed to a diffusion-reaction system, for one- and two-dimensional cases. On the other hand, an imposed singular vortex advects initially on/off concentration fields, in combination with diffusion and reaction. This problem is addressed analytically, by characterizing the boundary by a box-counting dimension and the concentration fields by a Hölder exponent, and numerically, by direct numerical simulations of the advection-diffusion-reaction equations. Second, the way the two sources hang together shows that, depending on the self-similar properties of the initial concentration fields, the vortex promotes the chemical activity close to its inner smoothed-out core or close to the outer region where the boundary starts to spiral. For all the considered situations, the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws.

Kinetics of the bosonic A+B-->0 reaction with on-site attractive interaction

We investigate kinetics of the uniformly driven bosonic A+B-->0 reaction with on-site attractive interaction in one dimension. In this model, n(i)(lambda) particles from a site with n(i) particles are driven to the right. When particles of opposite species occupy the same site, the reaction takes place instantaneously. Since n(i)(lambda)<n(i) for lambda<1, lambda controls the on-site attractive interaction between like particles. The lambda=0 case corresponds to the hard-core (HC) particle model. With equal initial densities of both species, we numerically confirm that the scaling behaviors of density and lengths are the same as those of the uniformly driven HC particle system. Especially the domain length l satisfies the power law l approximately t(2/3). The kinetics of the reaction is independent of lambda as long as lambda<1. The lambda -independent kinetics results from the lambda -independent collective motions of single species domains.

Continuously varying exponents in A+B-->0 reaction with long-ranged attractive interaction

We investigate kinetics of A+B-->0 reaction with long-range attractive interaction V(r) approximately -r(-2sigma) between A and B or with drift velocity v approximately r(-sigma) in one dimension, where r is the closest distance between A and B . It is analytically shown that dynamical exponents for density of particles (rho) and size of domains (l) continuously vary with sigma when sigma < sigma(c) = 1/2 , while that for the distance between adjacent opposite species (l(AB)) varies when sigma < sigma(c)AB = 7/6 . For sigma > sigma(c)AB diffusive motions dominate the kinetics. These anomalous behaviors with the two crossover values of sigma are supported by numerical simulations.

Anomalous kinetics of attractive A + B-->0 reactions

We investigate the kinetics of the A + B-->0 reaction with the attractive interaction between opposite species in one spatial dimension. The attractive interaction leads to isotropic diffusions inside segregated single species domains, and accelerates the reactions of opposite species at the domain boundaries. At equal initial densities of and , we analytically and numerically show that the density of particles (rho), the size of domains (l), the distance between the closest neighbor of same species (lAA), and the distance between adjacent opposite species (lAB) scale in time as rho approximately t(-1/3), lAA approximately t(1/3), and l approximately lAB approximately lAB(2/3), respectively. These dynamical exponents define critical behavior distinguished from the class of uniformly driven systems of hard-core particles.

Multispecies pair annihilation reactions

We consider diffusion-limited reactions A(i)+A(j)--> (1< or =i<j< or =q) in d space dimensions. For q>2 and d> or =2, we argue that the asymptotic density decay for such mutual annihilation processes with equal rates and initial densities is the same as for single-species pair annihilation A+A-->. In d=1, however, particle segregation occurs for all q< infinity. The total density decays according to a q dependent power law, rho(t) approximately t(-alpha(q)). Within a simplified version of the model alpha(q)=(q-1)/2q can be determined exactly. Our findings are supported through Monte Carlo simulations.

Reaction front structure in the diffusion-limited A+B model with initially randomized reactants

Subtle features of the reaction front formation in the A+B-->0 reaction are reported for the initially random and equal A+B reactant distribution. Three nonclassical parameters (initial linewidth, minimum, and maximum), for each interparticle gap and nearest neighbor distance distributions, are derived, as a function of time, using Monte Carlo simulations. These empirical front measures and their temporal scaling exponents are compared with the previously studied ones for the reactant interparticle distributions.

Critical branching-annihilating random walk of two species

The effect of blocking between different species occurring in one dimension is investigated here numerically in the case of particles following branching and annihilating random walk. It is shown that two-dimensional simulations confirm the field theoretical results with logarithmic corrections. In one dimension, however, if particles exhibit hard core interaction I confirm the very recent predictions of Kwon et al. [Phys. Rev. Lett. 85, 1682 (2000)] that there are two different universality classes depending on the spatial symmetry of the offspring production characterized by beta(S)=0.5 and beta(A)=2. Elaborate analysis of simulation data shows that the order parameter exponent beta does not depend on initial conditions or on diffusion rates of species but strong correction to scaling is observed. By systematic numerical simulations the critical point properties have been explored and initial condition dependence of the dynamical exponents Z and alpha is shown. In the case of a random initial state the particle-density decay at the critical point follows the t(-1/4) law with logarithmic corrections with two offsprings.