Abstract
The Becker-Doring equations serve as a model for the nucleation of a new thermodynamic phase in a first-order phase transformation. This corresponds to the case when the total density of monomers exceeds a critical value and the excess density is contained in larger and larger clusters as time proceeds. It has been derived in Penrose [J. Stat. Phys. 89:1/2 (1997), 305-320] and Niethammer [J. Nonlin. Sci. 13:1 (2003), 115-155] that the evolution of these large clusters can on a certain large time scale be described by a nonlocal transport equation coupled with the constraint that the total volume of new phase is conserved. For specific coefficients this equation is well known as a classical mean-field model for coarsening. In the present paper we consider the regime of small excess density on a large time scale, but not as large as in Penrose (1997) or Niethammer (2003). We show rigorously that the leading order dynamics are governed by another variant of the classical mean-field model in which total mass is preserved.
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Niethammer, B. A Scaling Limit of the Becker-Döring Equations in the Regime of Small Excess Density. J Nonlinear Sci 14, 453–468 (2004). https://doi.org/10.1007/s00332-004-0638-5
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DOI: https://doi.org/10.1007/s00332-004-0638-5