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# Communications in Mathematical Sciences

## Volume 15 (2017)

### Number 5

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

Pages: 1353 – 1384

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n5.a7

#### Authors

#### 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–Döring model (BD) that describes discrete-sized clusters through an infinite set of ordinary differential equations. The second one, the Lifshitz–Lyozov equation (LS) that is a transport partial differential equation on the continuous half-line $x \in (0 , + \infty)$. We introduce a scaling parameter $\varepsilon \gt 0$, which accounts for the grid size of the state space in the BD model, and recover the LS model in the limit $\varepsilon \to 0$. 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.

#### Keywords

Becker–Döring system, Lifshitz–Slyozov equation, boundary value for transport equation, quasi-steady state approximation, hydrodynamic limit

#### 2010 Mathematics Subject Classification

34E13, 35F31, 82C26, 82C70

The authors thank the anonymous referees that helped us to significatively improve the paper, in particular to obtain global in time results, and to clarify the topology. EH thanks the financial support of CAPES/IMPA (Brazil) and the Departamento de Matemática, Universidade Federal de Campina Grande (Paraiba, Brasil) where the author stayed when participated to this work.

Paper received on 29 May 2016.