Solving the Low Dimensional Smoluchowski Equation with a Singular Value Basis Set

Gregory E. Scott, University of Illinois at Urbana-Champaign
Martin Gruebele, University of Illinois at Urbana-Champaign

Article comments

This is the pre-peer reviewed version of the following article: Solving the Low Dimensional Smoluchowski Equation with a Singular Value Basis Set, Gregory Scott and Martin Gruebele, Journal of Computational Chemistry, 31, Copyright © 2010 Wiley-Blackwell., which has been published in final form at http://dx.doi.org/10.1002/jcc.21535.

NOTE: At the time of publication, the author Gregory Scott was not yet affiliated with Cal Poly.

Abstract

Reaction kinetics on free energy surfaces with small activation barriers can be computed directly with the Smoluchowski equation. The procedure is computationally expensive even in a few dimensions. We present a propagation method that considerably reduces computational time for a particular class of problems: when the free energy surface suddenly switches by a small amount, and the probability distribution relaxes to a new equilibrium value. This case describes relaxation experiments. To achieve efficient solution, we expand the density matrix in a basis set obtained by singular value decomposition of equilibrium density matrices. Grid size during propagation is reduced from (100–1000)^N to (2–4)^N in N dimensions. Although the scaling with N is not improved, the smaller basis set nonetheless yields a significant speed up for low-dimensional calculations. To demonstrate the practicality of our method, we couple Smoluchowsi dynamics with a genetic algorithm to search for free energy surfaces compatible with the multiprobe thermodynamics and temperature jump experiment reported for the protein α₃D.