Stationary Kolmogorov solutions of the Smoluchowski aggregation equation with a source term

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.

Cluster–cluster aggregation as an analogue of a turbulent cascade: Kolmogorov phenomenology, scaling laws and the breakdown of self-similarity

“…arguments. Strictly speaking, the Zakharov transformation technology of [10,26] only works for homogeneous kernels which leaves us with the question of what to do with the logarithm in eq. (87).…”

“…Therefore, it is correctly predicted by mean field theory. The non-renormalization of γ 2 by diffusive fluctuations can be explained by mass conservation or, more precisely by constancy of the average flux of mass in the mass space, see [10] for more details. Here, we simply wish to point out that the exact answers for γ 1 and γ 2 establish multiscaling non-perturbatively: the points (0, 0), (1, γ 1 ) and (2, γ 2 ) do not lie on the same straight line.…”

“…From the exact solution of the two point correlation function, the exponents governing the single site mass distribution was found for arbitrary dimensions [9]. The single site mass distribution was solved for mass dependent aggregation kernel using Zakharov transformation [10] and using the Smoluchowski approximation in the context of submonolayer epitaxial thin film growth [11]. Some other contexts in which the model has been studied include granular bead packs [12], nonequilibrium phase transitions [13,14] and self organized criticality [15].…”

“…After application of the Zakharov transformations described in [10], the stationary RSE, eq. (88), it becomes…”

“…The stationary state of this equation is best studied using the method of Zakharov transformations as detailed in [10,26] …”