Fluctuation theory of the coupling between diffusion and reaction in 1D: coalescence with point reactivities

Fluctuation theory of the one-dimensional coalescence of point particles with finite intrinsic reactivity, k0, is presented. For irreversible coalescence, A + A → A, a closed nonclassic rate equation is derived. For reversible coalescence, A + A ⇌ A, the kinetics of relaxation to equilibrium are discussed. For steady states maintained by an external source, S → A, an implicit relation between the input rate and the concentrations is found. An analytic model developed by Ben-Avraham, Burschka and Doering (BBD) [J. Stat. Phys. 60 (1990) 695] is used to test the approximations of the fluctuation theory for the limiting case of infinite intrinsic reactivity, k0 → ∞. Semiquantitative agreement is found between the fluctuation theory and the BBD model except for relaxation to equilibrium. The kinetic relationships between coalescence (A + A → A) and annihilation (A + A → 0) predicted by the fluctuation theory coincide with those from the model analytic kinetics.