Fluctuation-induced spreading of size distribution in condensation kinetics

One of the major results of condensation theory is the time independence of the size distribution shape (in terms of a certain invariant size) at the stage of regular growth of particles. This property follows directly from the simplified Zeldovich equation in the continuous form, where the fluctuation term is neglected. We show that the time invariance is broken by the fluctuation-induced spreading of the size spectrum. We first analyze the linear kinetic equations for the distributions p_i(t) with the growth rates of the form i^α. Exact solutions demonstrate the increase in dispersion with time as √t at α =0 and the time-independent dispersion at α =1. From the asymptotic analysis of the continuous Zeldovich equation with fractional α, it is shown that the distribution spreading always occurs at α <1/2. We then study the general case of homogeneous condensation in an open system with pumping. Asymptotical solutions for the size distribution have the form of a diffusionlike Gaussian. In the case of constant material influx, the spectrum width increases with mean size z as √z irrespective of α. We present a diagram of different growth scenarios and show that the time spreading occurs in the majority of condensing systems. Some numerical estimates for the effect of spectrum spreading are also presented.