Abstract
In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation with a source term for arbitrary homogeneous coagulation kernel. The resulting power-law mass distributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. They are valid for masses much larger than the characteristic mass of the source. We derive a "locality criterion," expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissible solution. Whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the Kolmogorov spectrum. As an example, we compute the exact stationary state for the family of kernels, K(zeta) ( m(1), m(2) )= ( m(1) m(2) )(zeta/2) which includes both gelling and nongelling cases, reproducing the known solution in the case zeta=0. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.
Citations
Jun 7, 2012·The Journal of Chemical Physics·P L Krapivsky, Colm Connaughton
Aug 28, 2015·Molecular Membrane Biology·Mirella V KolevaAnthony I Magee
Jan 9, 2016·Theoretical Population Biology·M A IrvineM J Keeling
Mar 16, 2007·Physical Review Letters·Colm ConnaughtonOleg Zaboronski
Jul 23, 2008·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·E Ben-Naim, P L Krapivsky
Apr 7, 2010·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Colm Connaughton, Alan C Newell
Apr 7, 2010·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Colm Connaughton, P L Krapivsky
Aug 27, 2011·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Robin C BallOleg Zaboronski
Nov 16, 2013·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Md Kamrul HassanNabila Islam
Dec 15, 2015·PloS One·Quentin VagnePierre Sens
Mar 20, 2018·Physical Review. E·Colm ConnaughtonOleg Zaboronski
Jun 17, 2018·Physical Review. E·Izumi SaitoTakeshi Watanabe
Feb 27, 2018·Physical Review Letters·Quentin Vagne, Pierre Sens
Feb 23, 2021·PLoS Computational Biology·Daniel J A DerrickMarcus John Tindall
Dec 12, 2012·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Gabriel Seiden, Victor Steinberg
Dec 12, 2012·Physical Review Letters·Robin C BallOleg Zaboronski
Nov 13, 2008·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Colm ConnaughtonOleg Zaboronski
Aug 16, 2013·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Peter P JonesColm Connaughton