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 for arbitrary homogeneous kernel. The resulting massdistributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. 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 admissiblesolution. 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 non-gelling cases, reproducing the known solution in the case $\zeta =0$. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.

This article is an expanded version of a talk given at IHP workshop "Dynamics, Growth and Singularities of Continuous Media", Paris July 2003. Updated 01/04/04. Revised version with additional discussion, references added, several typographical errors corrected. Revised version accepted for publication by Phys. Rev. E

References in corpus (5)

• Scaling Theory and Exactly Solved Models In the Kinetics of Irreversible Aggregation
• Influence of Island Diffusion on Submonolayer Epitaxial Growth
• Logarithmic Clustering in Submonolayer Epitaxial Growth
• Kang-Redner Anomaly in Cluster-Cluster Aggregation
• Stochastic Aggregation of Point Particles Revisited

Cited by in corpus (24)

• Pattern formation by turbulent cascades
• Cluster-Cluster Aggregation as an Analogue of a Turbulent Cascade : Kolmogorov Phenomenology, Scaling Laws and the Breakdown of self-similarity
• Instantaneous Gelation in Smoluchowski's Coagulation Equation Revisited
• Stationary mass distribution and nonlocality in models of coalescence and shattering
• Numerical Solutions of the Isotropic 3-Wave Kinetic Equation
• Phase Transition with Non-Thermodynamic States in Reversible Polymerization
• Dynamical Scaling and the Finite Capacity Anomaly in 3-Wave Turbulence
• Coalescence of particles by differential sedimentation
• Driven Brownian coagulation of polymers
• Aggregation-Fragmentation Processes and Wave Kinetics
• Constant flux relation for driven dissipative systems
• Stochastic Models of Vesicular Sorting in Cellular Organelles
• Universality properties of steady driven coagulation with collisional evaporation
• Constant Flux Relation for diffusion limited cluster-cluster aggregation
• A Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation
• Emergence of fractal in aggregation with stochastic self-replication
• Dyadic Cantor set and its kinetic and stochastic counterpart
• Hydrodynamic equations for space-inhomogeneous aggregating fluids with first-principle kinetic coefficients
• Mass Exchange Processes with Input
• Gelation in input-driven aggregation
• A nonlinear least squares method for the inverse droplet coagulation problem
• Aggregation Driven by a Localized Source
• Nonlocal interactions in coagulating particle systems
• Scaling properties of one-dimensional cluster-cluster aggregation with Levy diffusion