Stochastic Coagulation Equation and Validity of the Statistical Coagulation Equation.

Abstract

In the study of the origin of the solar system, many works on the coagulation process (i.e., the growth of dust grains and of planetesimals) have been made by using the coagulation equation which governs the time variation of the size distribution of “particles” due to collisional coalescence. As is conjectured by many authors, however, the coagulation equation itself may have a serious defect in some cases and, hence, cannot describe precisely the coagulation process; the essence of the defect lies on a fact that the coagulation equation inevitably creates particles with infinite mass. The purpose of the present study is to examine the conditions under which the coagulation equation can be regarded to be valid. First, on the basis of the stochastic viewpoint, we derived a fundamental equation (called `the stochastic coagulation equation') which describes exactly the coagulation process in a finite particle system. Then, investigating the properties of the stochastic coagulation equation, we found that it reduces to the ordinary coagulation equation if the correlation between the numbers of particles with masses i and j (i.e., (ninj) - (ni) (nj)) is negligibly small compared with (ni) (nj) (ni being the number of particles with mass i and brackets mean the ensemble average) and if the total particle number is sufficiently large. Furthermore, we solved the stochastic coagulation equation for the three special cases where the coalescence rate Aij is given by 1, i + j, and i × j and obtained its analytic solutions. Using the solutions of the latter two cases, we found that the ordinary coagulation equation, at least, can approximately describe the number of particles of which mass is much smaller than the geometrical mean of the smallest mass and the total mass of the particle system.