Diffusion-limited coalescence with finite reaction rates in one dimension

Steady state of a two-species annihilation process with separated reactants

We describe the steady state of the annihilation process of a one-dimensional system of two initially separated reactants A and B. The parameters that define the dynamical behavior of the system are the diffusion constant, the reaction rate, and the deposition rate. Depending on the ratio between those parameters, the system exhibits a crossover between a diffusion-limited (DL) regime and a reaction-limited (RL) regime. We found that a key quantity to describe the reaction process in the system is the probability p(x_{A},x_{B}) to find the rightmost A (RMA) particle and the leftmost B (LMB) particle at the positions x_{A} and x_{B}, respectively. The statistical behavior of the system in both regimes is described using the density of particles, the gap length distribution x_{B}-x_{A}, the marginal probabilities p_{A}(x_{A}) and p_{B}(x_{B}), and the reaction kernel. For both regimes, this kernel can be approximated by using p(x_{A},x_{B}). We found an excellent agreement between the numerical and analytical results for all calculated quantities despite the reaction process being quite different in both regimes. In the DL regime, the reaction kernel can be approximated by the probability to find the RMA and LMB particles in adjacent sites. In the RL regime, the kernel depends on the marginal probabilities p_{A}(x_{A}) and p_{B}(x_{B}).

Analytic solutions for some reaction-diffusion scenarios

Motivated currently by the problem of coalescence of receptor clusters in mast cells in the general subject of immune reactions, and formerly by the investigation of exciton trapping and sensitized luminescence in molecular systems and aggregates, we present analytic expressions for survival probabilities of moving entities undergoing diffusion and reaction on encounter. Results we provide cover several novel situations in simple 1-d systems as well as higher-dimensional counterparts along with a useful compendium of such expressions in chemical physics and allied fields. We also emphasize the importance of the relationship of discrete sink term analysis to continuum boundary condition studies.

Measurement of a reaction-diffusion crossover in exciton-exciton recombination inside carbon nanotubes using femtosecond optical absorption

Exciton-exciton recombination in isolated semiconducting single-walled carbon nanotubes was studied using femtosecond transient absorption. Under sufficient excitation to saturate the optical absorption, we observed an abrupt transition between reaction- and diffusion-limited kinetics, arising from reactions between incoherent localized excitons with a finite probability of ~0.2 per encounter. This represents the first experimental observation of a crossover between classical and critical kinetics in a 1D coalescing random walk, which is a paradigm for the study of nonequilibrium systems.

Mean-field approximation for spacing distribution functions in classical systems

We propose a mean-field method to calculate approximately the spacing distribution functions p((n))(s) in one-dimensional classical many-particle systems. We compare our method with two other commonly used methods, the independent interval approximation and the extended Wigner surmise. In our mean-field approach, p((n))(s) is calculated from a set of Langevin equations, which are decoupled by using a mean-field approximation. We find that in spite of its simplicity, the mean-field approximation provides good results in several systems. We offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system. The physical interpretation of each method is also discussed.

Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment, and deposition

Monolayer cluster growth in far-from-equilibrium systems is investigated by applying simulation and analytic techniques to minimal hard core particle (exclusion) models. The first model (I), for postdeposition coarsening dynamics, contains mechanisms of diffusion, attachment, and slow activated detachment (at rate epsilon<<1 ) of particles on a line. Simulation shows three successive regimes of cluster growth: fast attachment of isolated particles; detachment allowing further ( epsilont )(1/3) coarsening of average cluster size; and t(-1/2) approach to a saturation size varying as epsilon(-1/2) . Model II generalizes the first one in having an additional mechanism of particle deposition into cluster gaps, suppressed for the smallest gaps. This model exhibits early rapid filling, leading to slowing deposition due to the increasing scarcity of deposition sites, and then continued power law [ ( epsilont )(1/2) ] cluster size coarsening through the redistribution allowed by slow detachment. The basic ( epsilont )(1/3) domain growth laws and epsilon(-1/2) saturation in model I are explained by a simple scaling picture involving the time for a particle to detach and diffuse to the next cluster. A second, fuller approach is presented that employs a mapping of cluster configurations to a column picture and an approximate factorization of the cluster configuration probability within the resulting master equation. This allows, through the steady state solution of the corresponding equation for a cluster probability generating function, quantitative results for the saturation of model I in excellent agreement with the simulation results. For model II, it provides a one-variable scaling function solution for the coarsening probability distribution, and in particular quantitative agreement with the cluster length scaling and its amplitude.