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Understanding the slowest relaxations of complex systems, such as relaxationof glass-forming materials, diffusion in nanoclusters, and folding ofbiomolecules, is important for physics, chemistry, and biology. For a kineticsystem, the relaxation modes are determined by diagonalizing its transitionrate matrix. However, for realistic systems of interest, numericaldiagonalization, as well as extracting physical understanding from thediagonalization results, is difficult due to the high dimensionality. Here, wedevelop an alternative and generally applicable method of extracting thelong-time scale relaxation dynamics by combining the metabasin analysis ofOkushima et al. [Phys. Rev. E 80, 036112 (2009)] and a Jacobi method. We testthe method on a illustrative model of a four-funnel model, for which we obtaina renormalized kinematic equation of much lower dimension sufficient fordetermining slow relaxation modes precisely. The method is successfully appliedto the vacancy transport problem in ionic nanoparticles [Niiyama et al. Chem.Phys. Lett. 654, 52 (2016)], allowing a clear physical interpretation that thefinal relaxation consists of two successive, characteristic processes.

Slowest kinetic modes revealed by metabasin renormalization