Quasi steady state approximation of the small clusters in Becker–Döring equations leads to boundary conditions in the Lifshitz–Slyozov limit - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Communications in Mathematical Sciences Year : 2017

Quasi steady state approximation of the small clusters in Becker–Döring equations leads to boundary conditions in the Lifshitz–Slyozov limit

Abstract

The following paper addresses the connection between two classical models of phase transition phenomena describing different stages of clusters growth. The first one, the Becker-Doring model (BD) that describes discrete-sized clusters through an infinite set of ordinary differential equations. The second one, the Lifshitz-Slyozov equation (LS) that is a transport partial differential equation on the continuous half-line x is an element of (0, + infinity). We introduce a scaling parameter epsilon > 0, which accounts for the grid size of the state space in the BD model, and recover the LS model in the limit epsilon -> 0. The connection has been already proven in the context of outgoing characteristic at the boundary x = 0 for the LS model when small clusters tend to shrink. The main novelty of this work resides in a new estimate on the growth of small clusters, which behave at a fast time scale. Through a rigorous quasi steady state approximation, we derive boundary conditions for the incoming characteristic case, when small clusters tend to grow.
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hal-01608844 , version 1 (26-05-2020)
hal-01608844 , version 2 (14-01-2021)

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Julien Deschamps, Erwan Hingant, Romain Yvinec. Quasi steady state approximation of the small clusters in Becker–Döring equations leads to boundary conditions in the Lifshitz–Slyozov limit. Communications in Mathematical Sciences, 2017, 15 (5), pp.1353-1384. ⟨10.4310/CMS.2017.v15.n5.a7⟩. ⟨hal-01608844v2⟩
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