Model reduction in Smoluchowski-type equations

Scaling and the Smoluchowski equations.

The Smoluchowski equations, which describe coalescence growth, take into account combination reactions between a j-mer and a k-mer to form a (j+k)-mer, but not breakup of larger clusters to smaller ones. All combination reactions are assumed to be second order, with rate constants K(jk). The K(jk) are said to scale if K(lambda j,gamma k) = lambda(mu)gamma(nu)K(jk) for j < or = k. It can then be...

Kinetic equations governing Smoluchowski dynamics in equilibrium.

We continue our study of the statistical properties of particles in equilibrium obeying Smoluchowski dynamics. We show that the system is governed by a kinetic equation of the memory function form and that the memory function is given by one of the self-energies available via perturbation theory as introduced in previous work. We determine the memory function explicitly to second order in an ex...

Asymptotics for Scaled Kramers-Smoluchowski Equations

We offer fairly simple and direct proofs of the asymptotics for the scaled Kramers-Smoluchowski equation in both one and higher dimensions. For the latter, we invoke the sharp asymptotic capacity asymptotics of Bovier–Eckhoff–Gayrard–Klein [B-E-G-K].

Matrix Equations and Model Reduction

Model order reduction methods for linear time invariant systems are reviewed in this lecture. The basic ideas of the methods, such as the Padé approximation method, the rational interpolation method, the modal truncation method, the standard balanced truncation method, and the balancing related methods, are presented. The numerical algorithms of implementing the methods are discussed. For the b...

On the Positivity of Solutions to the Smoluchowski Equations