Regular Article

Limited coagulation-diffusion dynamics in inflating spaces

¹ Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 08826, Republic of Korea
² Laboratoire de Physique et Chimie Théoriques, CNRS (UMR 7019), Université de Lorraine, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France
³ Department of Physics, University of Seoul, Seoul 02504, Republic of Korea

^a e-mail: jean-yves.fortin@univ-lorraine.fr

Received: 30 January 2020
Received in final form: 3 May 2020
Accepted: 11 August 2020
Published online: 14 September 2020

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 t^1∕2−β for β < 1∕2, logarithmic for β = 1∕2, and reaching a finite value for all 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.