Exact results for diffusion-limited reactions with synchronous dynamics

Quantum reaction-limited reaction-diffusion dynamics of annihilation processes

We investigate the quantum reaction-diffusion dynamics of fermionic particles which coherently hop in a one-dimensional lattice and undergo annihilation reactions. The latter are modelled as dissipative processes which involve losses of pairs 2A→∅, triplets 3A→∅, and quadruplets 4A→∅ of neighboring particles. When considering classical particles, the corresponding decay of their density in time follows an asymptotic power-law behavior. The associated exponent in one dimension is different from the mean-field prediction whenever diffusive mixing is not too strong and spatial correlations are relevant. This specifically applies to 2A→∅, while the mean-field power-law prediction just acquires a logarithmic correction for 3A→∅ and is exact for 4A→∅. A mean-field approach is also valid, for all the three processes, when the diffusive mixing is strong, i.e., in the so-called reaction-limited regime. Here we show that the picture is different for quantum systems. We consider the quantum reaction-limited regime and we show that for all the three processes power-law behavior beyond mean field is present as a consequence of quantum coherences, which are not related to space dimensionality. The decay in 3A→∅ is further, highly intricate, since the power-law behavior therein only appears within an intermediate time window, while at long times the density decay is not power law. Our results show that emergent critical behavior in quantum dynamics has a markedly different origin, based on quantum coherences, to that applying to classical critical phenomena, which is, instead, solely determined by the relevance of spatial correlations.

Reaction-Limited Quantum Reaction-Diffusion Dynamics

We consider the quantum nonequilibrium dynamics of systems where fermionic particles coherently hop on a one-dimensional lattice and are subject to dissipative processes analogous to those of classical reaction-diffusion models. Particles can either annihilate in pairs, A+A→0, or coagulate upon contact, A+A→A, and possibly also branch, A→A+A. In classical settings, the interplay between these processes and particle diffusion leads to critical dynamics as well as to absorbing-state phase transitions. Here, we analyze the impact of coherent hopping and of quantum superposition, focusing on the so-called reaction-limited regime. Here, spatial density fluctuations are quickly smoothed out due to fast hopping, which for classical systems is described by a mean-field approach. By exploiting the time-dependent generalized Gibbs ensemble method, we demonstrate that quantum coherence and destructive interference play a crucial role in these systems and are responsible for the emergence of locally protected dark states and collective behavior beyond mean field. This can manifest both at stationarity and during the relaxation dynamics. Our analytical results highlight fundamental differences between classical nonequilibrium dynamics and their quantum counterpart and show that quantum effects indeed change collective universal behavior.

Correlation functions for diffusion-limited annihilation, A+A-->0

The full hierarchy of multiple-point correlation functions for diffusion-limited annihilation, A+A-->0, is obtained analytically and explicitly, following the method of intervals. In the long-time asymptotic limit, the correlation functions of annihilation are identical to those of coalescence, A+A-->A, despite differences between the two models in other statistical measures, such as the interparticle distribution function.

Method of intervals for the study of diffusion-limited annihilation, A+A-->0

We introduce a method of intervals for the analysis of diffusion-limited annihilation, A+A-->0, on the line. The method leads to manageable diffusion equations whose interpretation is intuitively clear. As an example, we treat the following cases: (a) annihilation in the infinite line and in infinite (discrete) chains; (b) annihilation with input of single particles, adjacent particle pairs, and particle pairs separated by a given distance; (c) annihilation, A+A-->0, along with the birth reaction A-->3A, on finite rings, with and without diffusion.

Nonlinear reactive systems on a lattice viewed as Boolean dynamical systems

We present a stochastic, time-discrete Boolean model that mimics the mesoscopic dynamics of the desorption reactions A+A-->A+S and A+A-->S+S in a one-dimensional lattice. In the continuous-time limit, we derive a hierarchy of dynamical equations for the subset of moments involving contiguous lattice sites. The solution of the hierarchy allows to compute the exact dynamics of the mean coverage for both microscopic and coarse-grained initial conditions, which turn out to be different from the mean field predictions. The evolution equations for the mean coverage and the second-order moments are shown to be equivalent to those provided by a time-continuous master equation. The important role of higher-order fluctuations is brought out by the failure of a truncation scheme retaining only two-particle fluctuation correlations.

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.

Single species diffusion-influenced reaction A+A-->alphaA: validity of the smoluchowski approach

We investigate the single species diffusion-influenced reaction, A+A-->alphaA with a finite reactivity in all dimensions. The reaction model includes a pure coagulation (alpha=1) or a pure annihilation (alpha=0) model. We apply the hierarchical Smoluchowski approach to study the dimensional aspects of the fluctuation, reactivity, particle size, and alpha(0</=alpha</=1). The theoretical results are compared with those of the Monte Carlo simulations in one, two, and three regular dimensions. The simulation results reveal that the classical Smoluchowski approach is exact in the short time limit in all dimensions and in the long time limit in three dimensions. The hierarchical Smoluchowski approach is found to be numerically exact at all times in two and three dimensions. A numerical method to obtain the exact result of the annihilation for a finite reactivity in one dimension is presented. We also propose a quite accurate analytic solution for an arbitrary alpha for the infinite reactivity in one dimension.