Smoluchowski’s coagulation equation with injections: applications to clustering of nano-particles

  • 1 Ural Federal University, Ekaterinburg, Russian Federation
  • 2 The University of Manchester, Manchester, UK


The coagulation equation, proposed by Smoluchowski, has been widely used to describe aggregation phenomena in many fields of science since its inception. It considers a physical system of many particles, and each particle is characterized by a change of some nonnegative scalar quantity (e.g. volume). Assuming such a system to be spatially homogeneous and unbounded, considering only pairwise interactions and a balance relation of interacting particles, the Smoluchowski equation can be used to describe the evolution of a system of many particles. In this study, we construct an exact solution to this integro-differential equation containing an exponentially-decaying source term. This solution, in particular, describes the steady-state structural density of endosomes per cell carrying the nanoparticles (particles smaller than 100 nm). In addition, we derived an exact analytical solution to the unsteady-state coagulation equation in the Laplace transform image space. This solution can be inverted using numerical methods for Laplace transform inversion. For practical use, we derive an analytical solution to the non-stationary coagulation equation stitching the steady-state and initial distributions of structural density. Choosing the particular form of stitching functions, we demonstrate that the nonstationary solution evolves between the initial and steady-state distribution functions. Thus, analytical solutions obtained represent a general theoretical basis to describe the dynamics of cargo distributions in the endosomal network.



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