A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids

Abstract

A classical model used in the study of dynamics of polymeric liquids is the bead-spring chain representation of polymer molecules. The chain typically consists of a large number of beads and thus the state space V of its configuration, which is essentially the position of all the constituent beads, turns out to be high dimensional. The distribution function governing the configuration of a bead-spring chain undergoing shear flow is a Fokker-Planck equation on V. In this article, we present QMC methods for the approximate solution of the Fokker-Planck equation which are based on the time splitting technique to treat convection and diffusion separately. Convection is carried out by moving the particles along the characteristics and we apply the algorithms presented in [G. Venkiteswaran, M. Junk, QMC algorithms for diffusion equations in high dimensions, Math. Comput. Simul. 68 (2005) 23-41.] for diffusion. Altogether, we find that some of the QMC methods show reduced variance and thus slightly outperform standard MC.