Exact lattice calculations of dispersion coefficients in the presence of external fields and obstacles

Abstract

We present a study of the field-dependent dispersion coefficient of point-like particles in various 2D overdamped systems with obstructions (periodic, percolating, and trapping distributions of obstacles). These calculations profit from the synthesis of a newly proposed Monte Carlo algorithm--the first such algorithm that correctly reproduces the free dispersion coefficient in the presence of finite external fields--and an asymptotically exact calculation technique. The resulting method efficiently produces algebraic and numerical results without the need to actually perform Monte Carlo simulations. When compared to such simulations, our exact method features a negligible computational cost and exponentially small errors. Utilizing the power of this numerical method, we engage in comprehensive parametric analysis of several model systems, revealing very subtle effects that would otherwise be swamped by statistical errors or incur prohibitive computational costs. The unified framework presented here serves as a template for further applications of lattice random-walk models of biased diffusion.

A new set of Monte Carlo moves for lattice random-walk models of biased diffusion

“…These calculations were done using the algorithm presented in Ref. [18] (filled symbols) and the one proposed in this paper (empty symbols). (b) The obstacle is of size l × l, the lattice is of size 3l × 3l (with periodic boundary conditions), and the particle (not shown) is of size ξl × ξl.…”

“…The exact numerical method that we developed in collaboration with Dorfman [18] is not limited to the previous set of LMC moves. It can easily be modified to include other LMC moves, including diagonal moves.…”

“…It can easily be modified to include other LMC moves, including diagonal moves. Combining Dorfman's method [18,11,12] and our new LMC moves, we now have a way to compute the exact velocity and the exact diffusion coefficient of a particle in the presence of arbitrary driving field for any dimension d ≥ 2.…”

“…However, the resulting algorithm can only be used in Monte Carlo computer simulations because exact resolution methods [13,18] require a common time step for all jumps.…”

Modeling the separation of macromolecules: A review of current computer simulation methods

“…(20). In a non-vanishing field, D can be obtained with a numerically exact method based on the generalized Taylor-Aris dispersion theory [110].…”