Graph-based analysis of kinetics on multidimensional potential-energy surfaces

Abstract

The aim of this paper is twofold: one is to give a detailed description of an alternative graph-based analysis method, which we call saddle connectivity graph, for analyzing the global topography and the dynamical properties of many-dimensional potential-energy landscapes and the other is to give examples of applications of this method in the analysis of the kinetics of realistic systems. A Dijkstra-type shortest path algorithm is proposed to extract dynamically dominant transition pathways by kinetically defining transition costs. The applicability of this approach is first confirmed by an illustrative example of a low-dimensional random potential. We then show that a coarse-graining procedure tailored for saddle connectivity graphs can be used to obtain the kinetic properties of 13- and 38-atom Lennard-Jones clusters. The coarse-graining method not only reduces the complexity of the graphs, but also, with iterative use, reveals a self-similar hierarchical structure in these clusters. We also propose that the self-similarity is common to many-atom Lennard-Jones clusters.

Slowest kinetic modes revealed by metabasin renormalization

“…In this Rapid Communication, to overcome this difficulty, we develop an alternative renormalization method tailored for extracting the slow dynamics precisely, which is based upon metabasin analysis [36,37] and a variant of the Jacobi rotation method for matrix diagonalization. Through the accurate renormalization procedure, a slow kinetic equation is generated that can reproduce the slow relaxation modes precisely.…”

Visualization for the Physical Sciences

“…The system combines concepts from topological analysis, multi-dimensional scaling and graph layout to produce a visualization of the energy function that focuses on the energy minima (stable states) of the system and their relationships to each other. The closest related work is that of Flamm et al [FHSS07] and Okushima et al [ONIS09] on topological analysis in chemistry, and that of Gerber et al [GBPW10] on visual analysis for chemistry. Simulation data is 3D, time-dependent, uniform and the main challenge is feature detection where the features are states of energy minima and the relationships between these states.…”

Mean first passage times reconstruct the slowest relaxations in potential energy landscapes of nanoclusters

“…Recently, the dynamics of complex systems, such as the relaxation of glass-forming materials [1][2][3][4][5][6][7][8][9][10][11][12], the kinetics of biomolecules [13][14][15][16][17][18][19][20][21][22][23], and diffusion in nanoclusters [24][25][26][27][28][29][30], were studied in a unified way for Markov state models [31][32][33][34][35]. The slowest relaxation modes of these systems describe the bottleneck processes, and hence they are the most crucial, e.g., for understanding glass transitions and rapid formations of mixed crystals [36][37][38][39][40][41][42][43].…”

“…However, it is hard to understand why the eigenvectors are formed in the shapes of the numerical diagonalization results because the eigenvectors are quite highdimensional and complicated. To extract the essence of the relaxation properties of Markov state models, there have been many studies, such as lumping or renormalizing Markov state models [35-39, 42, 43], and applications of network algorithms, such as Dijkstra's shortest path algorithm [40]. Although there are many pioneering works concerning this problem [44][45][46][47][48][49][50][51][52], to the best of our knowledge, this problem has not yet been completely clarified.…”