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 x(t)/x(t)=alpha (1+alpha t/beta)(-1) with growth rate alpha and exponent beta, describing both algebraic and exponential (beta = infinity) growths. In the space continuous limit, the particle dynamics is known to be subdiffusive, with the diffusive length varying like t(1/2-beta) for beta < 1/2, logarithmic for <beta> = 1/2, and reaching a finite value for all beta > 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.