The invariant solution of Smoluchowski coagulation equation with homogeneous kernels based on one parameter group transformation

Abstract

The governing equation of self-preserving size distribution can be obtained by similarity transformation from the Smoluchowski coagulation equation. To deal with the variable upper limit of convolution, the one parameter group transformation is introduced to get the invariant form of the governing equation. However, whether the equation has an invariant solution depending on asymptotic property of the coagulation kernel at the lower-end boundary. For homogeneous kernel, it can be discussed with three classes, type I has no similarity solution, type II has similarity solution which should be analyzed, and type III has similarity solution. Under the existence of similarity solution, an improved iDNS algorithm is developed with fourth order Runge–Kutta method to get the invariant solution of Smoluchowski coagulation equation more efficiently. Based on the invariant solution, a new definition of entropy in statistical physics is proposed to distinguish the convergence of Smoluchowski coagulation equation mathematically. The new entropy is a simple function of algebraic mean volume and has a maximum. At long time, the entropy production approaches to zero, which means the convergence of solution for Smoluchowski coagulation equation mathematically, and the Cercignani conjecture is almost true for Smoluchowksi coagulatin equation.