Limited coagulation-diffusion dynamics in inflating spaces

Abstract

We consider the one-dimensional coagulation–diffusion problem on a dynamical expanding linear lattice, in which the effect of the coagulation process is balanced by the dilatation of the distance between particles. Distances x(t) follow the general law ẋ (t) ∕ x (t) = α (1 + αt ∕ β) -1 with growth rate α and exponent β, describing both algebraic and exponential (β = ∞) growths. In the space continuous limit, the particle dynamics is known to be subdiffusive, with the diffusive length varying like t1∕2−β for β 1∕2. We interpret and characterize quantitatively this phenomenon as a second order phase transition between an absorbing state and a localized state where particles are not reactive. We furthermore investigate the case when space is discrete and use a generating function method to solve the time differential equation associated with the survival probability. This model is then compared with models of growth on geometrically constrained two-dimensional domains, and with the theory of fractional diffusion in the subdiffusive case. We found in particular a duality relation between the diffusive lengths in the inflating space and the fractional theory.