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, 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.
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Fortin, JY., Durang, X. & Choi, M. Limited coagulation-diffusion dynamics in inflating spaces. Eur. Phys. J. B 93, 175 (2020). https://doi.org/10.1140/epjb/e2020-10058-9
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DOI: https://doi.org/10.1140/epjb/e2020-10058-9