In 1917, Marian von Smoluchowski presented a simple mathematical description of diffusioncontrolled reactions on the scale of individual molecules. His model postulated that a reaction would occur when two reactants were sufficiently close and, more specifically, presented a succinct relationship between the relative proximity of two reactants at the moment of reaction and the macroscopic react...

Theory and practice of reaction dynamics on two-dimensional potential energy surfaces is investigated. Nonequilibrium multidimensional barrier crossing, occurring when the initial density is located near the ridgeline separating reactants and products, is treated by solving the time-dependent Smoluchowski equation as a function of diffusion anisotropy. For a locally separable potential, and slo...

The convergence of stochastic particle systems representing physical advection, inflow, outflow and coagulation is considered. The problem is studied on a bounded spatial domain such that there is a general upper bound on the residence time of a particle. The laws on the appropriate Skorohod path space of the empirical measures of the particle systems are shown to be relatively compact. The pat...

We propose a kinetic model for the self-aggregation by amyloid proteins. By extending several well-known models for protein aggregation, the time evolution of aggregate concentrations containing r proteins, denoted c(r)(t), can be written in terms of generalized Smoluchowski kinetics. With this approach, we take into account all possible aggregation and fragmentation reactions involving cluster...

We have elucidated the effect of statistical asymmetry on the directed current in Smoluchowski-Feynman ratchets driven by chaotic noise. Based on the inhomogeneous Smoluchowski equation and its generalized version, we arrive at analytical expressions of the directed current that includes a source term. The source term indicates that statistical asymmetry can drive the system further away from t...

We prove global existence for a nonlinear Smoluchowski equation (a nonlinear FokkerPlanck equation) coupled with Navier-Stokes equations in 2d. The proof uses a deteriorating regularity estimate in the spirit of [5] (see also [1]) Key wordsNonlinear Fokker-Planck equations, Navier-Stokes equations, Smoluchowski equation, micro-macro interactions. AMS subject classification 35Q30, 82C31, 76A05.

We develop the analogy between self-gravitating Brownian particles and bacterial populations. In the high friction limit, the self-gravitating Brownian gas is described by the Smoluchowski-Poisson system. These equations can develop a self-similar collapse leading to a finite time singularity. Coincidentally, the Smoluchowski-Poisson system corresponds to a simplified version of the Keller-Sege...

We present a simulation algorithm that accurately propagates a molecule pair using large time steps without the need to invoke the full exact analytical solutions of the Smoluchowski diffusion equation. Because the proposed method only uses uniform and Gaussian random numbers, it allows for position updates that are two to three orders of magnitude faster than those of a corresponding scheme ba...