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

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.

Multichain models of conserved lattice gas

Conserved lattice-gas models in one dimension exhibit absorbing state phase transition (APT) with simple integer exponents β=1=ν=η, whereas the same on a ladder belong to directed percolation (DP) universality. We conjecture that additional stochasticity in particle transfer is a relevant perturbation and its presence on a ladder forces the APT to be in the DP class. To substantiate this we introduce a class of restricted conserved lattice-gas models on a multichain system (M×L square lattice with periodic boundary condition in both directions), where particles which have exactly one vacant neighbor are active and they move deterministically to the neighboring vacant site. We show that for odd number of chains, in the thermodynamic limit L→∞, these models exhibit APT at ρ_{c}=1/2(1+1/M) with β=1. On the other hand, for even-chain systems transition occurs at ρ_{c}=1/2 with β=1,2 for M=2,4, respectively, and β=3 for M≥6. We illustrate this unusual critical behavior analytically using a transfer-matrix method.

Coarsening and persistence in a one-dimensional system of orienting arrowheads: Domain-wall kinetics with A+B→0

We demonstrate the large-scale effects of the interplay between shape and hard-core interactions in a system with left- and right-pointing arrowheads <> on a line, with reorientation dynamics. This interplay leads to the formation of two types of domain walls, >< (A) and <> (B). The correlation length in the equilibrium state diverges exponentially with increasing arrowhead density, with an ordered state of like orientations arising in the limit. In this high-density limit, the A domain walls diffuse, while the B walls are static. In time, the approach to the ordered state is described by a coarsening process governed by the kinetics of domain-wall annihilation A+B→0, quite different from the A+A→0 kinetics pertinent to the Glauber-Ising model. The survival probability of a finite set of walls is shown to decay exponentially with time, in contrast to the power-law decay known for A+A→0. In the thermodynamic limit with a finite density of walls, coarsening as a function of time t is studied by simulation. While the number of walls falls as t^{-1/2}, the fraction of persistent arrowheads decays as t^{-θ} where θ is close to 1/4, quite different from the Ising value. The global persistence too has θ=1/4, as follows from a heuristic argument. In a generalization where the B walls diffuse slowly, θ varies continuously, increasing with increasing diffusion constant.

Multicritical absorbing phase transition in a class of exactly solvable models

We study diffusion of hard-core particles on a one-dimensional periodic lattice subjected to a constraint that the separation between any two consecutive particles does not increase beyond a fixed value n+1; an initial separation larger than n+1 can however decrease. These models undergo an absorbing state phase transition when the conserved particle density of the system falls below a critical threshold ρ_{c}=1/(n+1). We find that the ϕ_{k}, the density of 0-clusters (0 representing vacancies) of size 0≤k<n, vanish at the transition point along with activity density ρ_{a}. The steady state of these models can be written in matrix product form to obtain analytically the static exponents β_{k}=n-k and ν=1=η corresponding to each ϕ_{k}. We also show from numerical simulations that, starting from a natural condition, ϕ_{k}(t)s decay as t^{-α_{k}} with α_{k}=(n-k)/2 even though other dynamic exponents ν_{t}=2=z are independent of k; this ensures the validity of scaling laws β=αν_{t} and ν_{t}=zν.

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.

Absorbing phase transition in a conserved lattice gas model with next-nearest-neighbor hopping in one dimension

The absorbing phase transition of the modified conserved lattice gas (m-CLG) model was investigated in one dimension. The m-CLG model was modified from the conserved lattice gas (CLG) model in such a way that each active particle hops to one of the nearest-neighbor and next-nearest-neighbor empty sites. The order parameter exponent, the dynamic exponent, and the correlation length exponent were estimated from the power-law behavior and finite-size scaling of the active particle densities. The exponents were found to differ considerably from those of the ordinary CLG model and were also distinct from those of the Manna model, suggesting that next-nearest-neighbor hopping is a relevant factor that alters the critical behavior in the one-dimensional CLG model.

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.

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.