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

Efficiency scaling of non-coherent upconversion in a one-dimensional model system

A very promising approach to obtain efficient upconversion of light is the use of triplet-triplet annihilation of excitations in molecular systems. In real materials, besides upconversion, many other physical processes take place--fluorescence, phosphorescence, non-radiative decay, annihilation, diffusion--and compete with upconversion. The main objective of this work is to design a proof of principle model that can be used to shed light on the interplay between these processes. Ultimately, we want to establish general principles that may guide experimentalists toward the design of solid state materials with maximum efficiency. Here we show, in a one-dimensional model system, that upconversion can be optimized by varying the ratio between the two molecular species used in triplet-triplet-annihilation based upconversion systems, even in the presence of undesired losses through phosphorescence, non-radiative decay, or annihilation. We derive scaling laws for this ratio and for the maximum efficiency of upconversion, as a function of the diffusion rate J, as well as of the creation and of the decay rate of the excitations.

Coagulation reactions in low dimensions: Revisiting subdiffusive A+A reactions in one dimension

We present a theory for the coagulation reaction A+A-->A for particles moving subdiffusively in one dimension. Our theory is tested against numerical simulations of the concentration of A particles as a function of time ("anomalous kinetics") and of the interparticle distribution function as a function of interparticle distance and time. We find that the theory captures the correct behavior asymptotically and also at early times, and that it does so whether the particles are nearly diffusive or very subdiffusive. We find that, as in the normal diffusion problem, an interparticle gap responsible for the anomalous kinetics develops and grows with time. This corrects an earlier claim to the contrary on our part.

Multiscaling of correlation functions in single species reaction-diffusion systems

We derive the multi-scaling of probability distributions of multi-particle configurations for the binary reaction-diffusion system in A + A --> O in d < or =2 and for the ternary system in 3A --> O in d=1. For the binary reaction we find that the probability Pt(N, Delta V) of finding N particles in a fixed volume element Delta V at time t decays in the limit of large time as (ln t/t)N(ln t)-N(N-1)/2 for d=2 and t-Nd/2 t-N(N-1)epsilon/4+O(epsilon2) for d<2. Here epsilon=2-d. For the ternary reaction in one dimension we find that Pt(N, delta V) approximately (ln t/t)N/2(ln t)-N(N-1)(N-2)/6 . The principal tool of our study is the dynamical renormalization group. We compare predictions of epsilon expansions for Pt(N, Delta V) for a binary reaction in one dimension against the exact known results. We conclude that the epsilon corrections of order two and higher are absent in the previous answer for Pt(N, Delta V) for N=1, 2, 3, 4. Furthermore, we conjecture the absence of epsilon2 corrections for all values of N.

Reaction front in an A+B-->C reaction-subdiffusion process

We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.

Long range hops and the pair annihilation reaction A+A-->0: renormalization group and simulation

A simple example of a nonequilibrium system for which fluctuations are important is a system of particles which diffuse and may annihilate in pairs on contact. The renormalization group can be used to calculate the time dependence of the density of particles, and provides both an exact value for the exponent governing the decay of particles and an epsilon expansion for the amplitude of this power law. When the diffusion is anomalous, as when the particles perform Lévy flights, the critical dimension depends continuously on the control parameter for the Lévy distribution. The epsilon expansion can then become an expansion in a small parameter. We present the renormalization group calculation and compare these results with those of a simulation.

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.