Limited coagulation-diffusion dynamics in inflating spaces
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THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Limited coagulation-diffusion dynamics in inflating spaces Jean-Yves Fortin 1,2,a , Xavier Durang 3 , and MooYoung Choi 1 1
2
3
Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 08826, Republic of Korea Laboratoire de Physique et Chimie Th´eoriques, CNRS (UMR 7019), Universit´e de Lorraine, BP 70239, 54506 Vandoeuvre-l`es-Nancy Cedex, France Department of Physics, University of Seoul, Seoul 02504, Republic of Korea Received 30 January 2020 / Received in final form 3 May 2020 / Accepted 11 August 2020 Published online 14 September 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 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 x(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.
1 Introduction Cooperative effects in low-dimensional systems with strongly interacting particles present a rich variety of critical properties in the regime out of equilibrium. Model examples of such systems are provided by one dimensional reaction-diffusion processes [1–3], which are relevant to a number of non-equilibrium physical cases, such as excitons in polymer chains TMMC = (CH3 )4 N(MnCl3 ) [4] or relaxation of photoexcitations in carbon-nanotubes [5]. The low dimensionality induces strong fluctuations that dominate the kinetics, and invalidates all descriptions based on mean-field theories below the upper critical dimension dc = 2. Exact results are therefore of importance in describing the dynamical behavior with precision by taking into account all the correlations. Here, we consider one of these coagulation–diffusion processes on a one-dimensional lattice with discrete sites of elementary size a. The Markov dynamics is defined by the particle elementary moves at the same rate τ −1 : A + ∅ → ∅ + A or ∅ + A → A + ∅ for the diffusion pr
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