Aggregate size distributions in migration driven growth models
The kinetics of aggregate growth through reversible migrations between any two aggregates is studied. We propose a simple model with the symmetrical migration rate kernel <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$K(k;j)\propto (kj)^\upsilon$</EquationSource> </InlineEquation> at which the monomers migrate from the aggregates of size k to those of size j. The results show that for the <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\upsilon \leq 3/2$</EquationSource> </InlineEquation> case, the aggregate size distribution approaches a conventional scaling form; moreover, the typical aggregate size grows as <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$t^{1 / (3 - 2\upsilon )}$</EquationSource> </InlineEquation> in the <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$ \upsilon > 3/2$</EquationSource> </InlineEquation> case and as <InlineEquation ID="Equ5"> <EquationSource Format="TEX">$\exp(C_1 t)$</EquationSource> </InlineEquation> in the <InlineEquation ID="Equ6"> <EquationSource Format="TEX">$\upsilon=3/2$</EquationSource> </InlineEquation> case. We also investigate another simple model with the asymmetrical rate kernel <InlineEquation ID="Equ7"> <EquationSource Format="TEX">$K(k;j)\propto k^\mu j^\nu$</EquationSource> </InlineEquation> (<InlineEquation ID="Equ8"> <EquationSource Format="TEX">$\mu \neq \nu$</EquationSource> </InlineEquation>), which exhibits some scaling properties quite different from the symmetrical one. The aggregate size distribution satisfies the conventional scaling form only in the case of <InlineEquation ID="Equ9"> <EquationSource Format="TEX">$\mu > \nu$</EquationSource> </InlineEquation> and <InlineEquation ID="Equ10"> <EquationSource Format="TEX">$\mu + \nu > 2$</EquationSource> </InlineEquation>, and the typical aggregate size grows as <InlineEquation ID="Equ11"> <EquationSource Format="TEX">$t^{2-\mu-\nu}$</EquationSource> </InlineEquation>. Copyright Springer-Verlag Berlin/Heidelberg 2003
