A + B → 0 Reaction with Different Initial Patterns

Kinetics of contact processes under segregation

The kinetics of contact processes are determined by the interplay among local mass transfer mechanisms, spatial heterogeneity, and segregation. Determining the macroscopic behavior of a wide variety of phenomena across the disciplines requires linking reaction times to the statistical properties of spatially fluctuating quantities. We formulate the dynamics of advected agents interacting with segregated immobile components in terms of a chemical continuous-time random walk. The inter-reaction times result from the first-passage times of mobile species to and across reactive regions, and available immobile reactants undergo a restart procedure. Segregation leads to memory effects and enhances the role of concentration fluctuations in the large-scale dynamics.

Survival and confinement under quenched disorder

We study the survival and confinement of random walkers under quenched disorder characterized by spatially-varying waiting times and decay rates. Spatial heterogeneity and segregation lead to a dynamic coupling between transport and reaction, resulting in history-dependent dynamics exhibiting long survivals and confinement. The survival probability decays as a power law, in contrast to the classical exponential law for decay at a homogeneous rate. The mean squared displacement shows dimension-dependent subdiffusive growth followed by localization, with stronger confinement in higher dimensions.

Reaction efficiency effects on binary chemical reactions

We study the effect of the variation of reaction efficiency in binary reactions. We use the well-known A + B → 0 model, which has been extensively studied in the past. We perform simulations on this model where we vary the efficiency of reaction, i.e., when two particles meet they do not instantly react, as has been assumed in previous studies, but they react with a probability γ, where γ is in the range 0 < γ < 1. Our results show that at small γ values the system is reaction limited, but as γ increases it crosses over to a diffusion limited behavior. At early times, for small γ values, the particle density falls slower than for larger γ values. This fall-off goes over a crossover point, around the value of γ = 0.50 for high initial densities. Under a variety of conditions simulated, we find that the crossover point was dependent on the initial concentration but not on the lattice size. For intermediate and long times simulations, all γ values (in the depleted reciprocal density versus time plot) converge to the same behavior. These theoretical results are useful in models of epidemic reactions and epidemic spreading, where a contagion from one neighbor to the next is not always successful but proceeds with a certain probability, an analogous effect with the reaction probability examined in the current work.

Effects of velocity relaxation on the anomalous kinetics of a one-dimensional A+A-->Ø reaction

A one-dimensional A+A-->slashed circle reaction with alternating dichotomic velocities is investigated to study the effects of velocity relaxation on the anomalous kinetics of this reaction. While we keep the magnitudes constant, the particle velocities are allowed to change from one moving direction to the other with a relaxation time, beta. The kinetics of the reaction is studied for various relaxation times. Although the anomalous slower reaction rate is anticipated for the reaction in one-dimension, compared to the classical rate law, the rate is found to speed up at intermediate times for intermediate values of beta . The time evolution for the spatial distribution of particles is also discussed to elucidate the effects of the velocity relaxation times on the kinetics of the reaction.

On-lattice coalescence and annihilation of immobile reactants in loopless lattices and beyond

We study the behavior of the chemical reactions A+A-->A+S and A+A-->S+S (where the reactive species A and the inert species S are both assumed to be immobile) embedded on Bethe lattices of arbitrary coordination number z and on a two-dimensional (2D) square lattice. For the Bethe lattice case, exact solutions for the coverage in the A species in terms of the initial condition are obtained. In particular, our results hold for the important case of an infinite one-dimensional (1D) lattice (z=2). The method is based on an expansion in terms of conditional probabilities which exploits a Markovian property of these systems. Along the same lines, an approximate solution for the case of a 2D square lattice is developed. The effect of dilution in a random initial condition is discussed in detail, both for the lattice coverage and for the spatial distribution of reactants.

Finite-size effects of two-particle diffusion-limited reactions

We have studied the finite-size effects of two-particle diffusion-limited reaction A+B-->0, on a one-dimensional lattice using the Monte Carlo method. The density at a finite lattice follows a power law, C(t) approximately t(-x) below the crossover time, and shows an exponential decay above the crossover time. The crossover time depends on the lattice size and the bias field. We found second-order correction terms of the density decay as C(t) approximately t(-1/4)[1+O(t(-1/8))] for the isotropic diffusion of particles and C(t) approximately t(-1/3)[1+O(t(-1/24))] for the maximum drift. We proposed the scaling function of the density given as C(t) approximately L(-1/2)f(o)(t/L(2))+L(-3/4)f(1)(t/L(2)) for the isotropic diffusion and C(t) approximately L(-1/2)f(o)(t/L(3/2))+L(-9/16)f(1)(t/L(3/2)) for the maximum drift where f(o) and f(1) are scaling functions.