Kinetics of bimolecular reactions in condensed media: critical phenomena and microscopic self-organisation

Abstract

Starting from the analysis of the hierarchy of equations for many-point reactant densities involved in three kinds of basic bimolecular reactions, A + A + B, A + B + C, A + B + B, in condensed media, a review is given of a new class of self-organisation phenomena. Unlike the usual synergetic effects, these phenomena are characterised by the appearance of microscopic dynamical clustering of similar reactants, which, however, does not violate the macroscopic homogeneity of the system. The many-particle effects are described in terms of the correlation length and critical exponents in much the same way as is done in the theory of critical phenomena (phase transitions) developed in statistical physics. It is shown that microscopic self-organisation results in asymptotic decay laws for reactant densities which are unusual for standard physical and chemical kinetics. The corresponding reduction of reaction rate with time is due to the emergence, in the course of biomolecular reaction, of a non-Poisson fluctuation spectrum of reactant densities governing the time development of average quantities. The universal character of the newly discovered self-organisation phenomena has been demonstrated to occur not only in numerous kinds of diffusion-controlled reactions, but also for static reactions at low temperatures, including reactant accumulation, when there is a source creating them (e.g. irradiation) and a long-range (tunnelling) recombination of immobile donors and acceptors in crystals. The mathematical formalism developed is applied to the two-stage bimolecular processes using the Lotka and Lotka-Volterra models as examples. Their analysis has revealed that in this case the generally accepted viewpoint on the self-organisation phenomena fails. This review was received in its present form in June 1988. 'Everything should be done as simply as possible but not simpler.' A Einstein 'Entia non sunt multiplicanda sine necesitate.

Long-range and many-body effects in coagulation processes

“…Therefore the flow of the vertex functions with m > n is zero. Similarly, the minimum number of incoming legs in the Feynman diagrams, which is n = 2, and the minimum number of outgoing legs, which is m = 1, is inherited to all scales k. Now, from the fact that the number of legs cannot increase along the time arrow, it is readily deduced that the flow of (1,1) k vanishes, since there is no diagram of the form of Fig. 1(a).…”

“…1(c)]. We then turn tog (1,2) , whose flow ∂ τg (1,2) τ is a function ofg (2,2) τ andg (1,2) τ . Assuming that we know the fixed-point values ofg (m,n) for all m < n we can go on to treatg (m,n) successively for m = n,n − 1, .…”

“…To calculate (1,2) k (0,0;0,0;0,0) we consider the flow of the more general vertex function (1,2) k (0,0;−p,−ω;p,ω) . It is determined by the one-loop diagram Fig.…”

“…One observes that the Padé approximation works the better, i.e., converges for larger values of χ , the closer one approaches the critical dimension. Indeed, from perturbation theory one expects that near the critical dimension onlyg (2,2) andg (1,2) are important, so that the approximation should converge quickly. Performing the limit d c − d = → 0, we can make contact with a result from perturbation theory [16].…”

Reaction–diffusion fronts with inhomogeneous initial conditions

“…Most of the Liesegang pattern studies have been made in homogeneous ‡ While the reagents in Liesegang phenomena are chemicals, examples where the reagents are of more exotic nature (such as topological defects [19]) are also known.…”