Crossover between diffusion-limited and reaction-limited regimes in the coagulation–diffusion process

Abstract

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.

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

“…One could not complete a work on diffusion without a brief discussion of reaction-diffusion processes [185,[188][189][190]. Exactly solvable reaction-diffusion models consist largely of single species reactions in one dimension, e.g., variations of the coalescence process, A + A → A + S [191][192][193] and the annihilation process A + A → S + S [192,193], where A and S denote occupied and empty sites, respectively.…”

“…This is one of the rare cases where theoretical statistical mechanics can be compared with experiments and also shows that simple mean-field schemes are not enough. For an introduction see [188], and for up to date references see [190].…”

Binary lattice-gases of particles with soft exclusion: Exact phase diagrams for tree-like lattices

“…The Bethe lattice approach has been employed, among many other examples, for the analysis of the phase diagram of the Blume-Emery-Griffiths model [37], of modulated phases emerging in the Ising model with competing interactions [38], the Potts model [39], lattice models of glassy systems [40] and localisation transitions [41], as well as a phase behaviour of confined ionic liquids [42,43]. Bethe lattice approach was also used in the analytical studies of different diffusion-reaction processes [44][45][46], processes of random and cooperative sequential adsorption [47] or of the structural properties of branched polymers [48,49]. The accuracy of such an approximation has been recently accessed in the numerical analysis of the Blume-Capel model and it was shown that, somewhat surprisingly, it provides a very accurate estimate of the location of the demarkation curve between the ordered and disordered phases [50].…”

Binary lattice-gases of particles with soft exclusion: exact phase diagrams for tree-like lattices

“…The Bethe lattice approach has been employed, among many other examples, for the analysis of the phase diagram of the Blume-Emery-Griffiths model [37], of modulated phases emerging in the Ising model with competing interactions [38], the Potts model [39], lattice models of glassy systems [40] and localisation transitions [41], as well as a phase behaviour of confined ionic liquids [42,43]. Bethe lattice approach was also used in the analytical studies of different diffusion-reaction processes [44][45][46], processes of random and cooperative sequential adsorption [47] or of the structural properties of branched polymers [48,49]. The accuracy of such an approximation has been recently assessed in the numerical analysis of the Blume-Capel model and it was shown that, somewhat surprisingly, it provides a very accurate estimate of the location of the demarkation curve between the ordered and disordered phases [50].…”