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.
Crossover between diffusion-limited and reaction-limited regimes in the coagulation–diffusion process
The change from the diffusion-limited to the reaction-limited cooperative behaviour in reaction–diffusion systems is analysed by comparing the universal long-time behaviour of the coagulation–diffusion process on a chain and on the Bethe lattice. On a chain, this model is exactly solvable through the empty-interval method. This method can be extended to the Bethe lattice, in the ben-Avraham–Glasser approximation. On the Bethe lattice, the analysis of the Laplace-transformed time-dependent particle-density is analogous to the study of the stationary state, if a stochastic reset to a configuration of uncorrelated particles is added. In this stationary state logarithmic corrections to scaling are found, as expected for systems at the upper critical dimension. Analogous results hold true for the time-integrated particle-density. The crossover scaling functions and the associated effective exponents between the chain and the Bethe lattice are derived.
Concentration dependence of diffusion controlled processes among static traps
An effective medium theory of competition effects in diffusion controlled reactions is presented. We consider a random array of static spherical traps which react with a density field of another reactant. Using the complete absorption boundary condition to describe the reaction at the surfaces of the traps, we calculate self‐consistently the concentration dependent rate constant for the reaction and the diffusion constant of the density field in the steady state at all trap concentrations. The rate constant does not diverge at any physical packing fraction of the traps in contrast to the case of regular arrays of traps.