Singular self-preserving regimes of coagulation processes

The late stages of the time evolution of disperse systems when either coagulation alone governs the temporal changes of particle mass spectra or simultaneous condensation complicates the evolution process are studied under the assumption that the condensation efficiencies and coagulation kernels are homogeneous functions of the particle masses, with gamma and lambda being their homogeneity exponents, respectively. In considering the asymptotic behavior of the particle mass distributions the renormalization-group approach is applied to three types of coagulating systems: free coagulating systems in which coagulation alone is responsible for disperse particle growth; source-enhanced coagulating systems, where an external spacially uniform source permanently adds fresh small particles, with the particle production being a power function of time; and coagulating-condensing systems in which a condensation process accompanies the coagulation growth of disperse particles. The particle mass distributions of the form N-A(g,t)=A(t)psi(gB(t)) are shown to describe the asymptotic regimes of particle growth in all the three types of coagulating systems (g is the particle mass). The functions A(t) and B(t) are normally power functions of time whose power exponents are found for all possible regimes of coagulation and condensation as the functions of lambda and gamma. The equations for the universality function psi(x) are formulated. It is shown that in many cases psi(x)proportional tox(-sigma) (sigma>1) at small x, i.e., the particle mass distributions are singular. The power exponent sigma is expressed in terms of lambda and gamma. Two exactly soluble models illustrate the general theoretical consideration.