Hierarchical Numerical Solution of Smoluchowski Equations with Rough Potentials

Area of Science:

• Computational Chemistry
• Physical Chemistry
• Materials Science

Background:

• Simulating diffusive processes on complex potential energy surfaces is computationally challenging.
• Existing methods like mean first passage time (MFPT) can be numerically unstable or inefficient for rough potentials.

Purpose of the Study:

• To introduce a novel, efficient, and numerically robust algorithm for diffusive processes.
• To explore the fine- and coarse-grained structure of interaction potentials without topological restrictions.

Main Methods:

• Hierarchical Discrete Approximation (HDA) algorithm.
• Utilizes a hierarchical grid to adaptively capture potential roughness.
• Comparison with conventional Discrete Approximation (DA) and MFPT methods.

Main Results:

• HDA demonstrates significant reduction in computational time and grid points needed.
• The algorithm is accurate and numerically stable, even for potentials with numerous minima (107).
• HDA shows monotonic convergence to analytical results for highly rough potentials.

Conclusions:

• HDA offers an efficient and robust solution for simulating diffusive processes on complex energy landscapes.
• The hierarchical approach is scalable to multidimensional problems.
• HDA provides a stable and accurate alternative to existing simulation methods.