Understanding the dynamics of complex systems requires the investigation of their energy landscape. In particular, the flow of probability on such landscapes is a central feature in visualizing the time evolution of complex systems. To obtain such flows, and the concomitant stable states of the systems and the generalized barriers among them, the threshold algorithm has been developed. Here, we describe the methodology of this approach starting from the fundamental concepts in complex energy landscapes and present recent new developments, the threshold-minimization algorithm and the molecular dynamics threshold algorithm. For applications of these new algorithms, we draw on landscape studies of three disaccharide molecules: lactose, maltose, and sucrose.
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There are different types of complex systems, some of which will be “complex” on all time scales of practical interest, but will exhibit such an exponential relaxation at extremely long observation times (e.g., glasses like SiO2 where a crystalline ground state exists and is accessible, in principle), while other systems will never show a global exponential relaxation behavior (such as ideal spin glasses in the computer).
Due to the probabilistic nature of their definition, the numerical values of the characteristic times should only be treated as approximate, and in extreme cases only as order of magnitude estimates.
We ignore restrictions due to special relativity.
Semi-classical here refers to systems where we, e.g., can map the quantum system to a model spin system with spins localized on a discrete lattice, or where the Born-Oppenheimer approximation holds and the ionic/atomic degrees of freedom can be treated as essentially classical leading to a standard multi-minima energy landscape for the atom arrangements of the chemical system.
Such regions are often called “stable states.”24,107 Intuitively, it is clear what this means, but the precise definition of what constitutes a stable state varies to a certain extent; we are not going to discuss these subtleties here.
Examples of nested marginally ergodic regions on a landscape are, e.g., amorphous or glassy compounds where aging phenomena108–110 are observed.
In the beginning of the study of multi-minima energy landscapes, one often performed only a single gradient-based deterministic minimization from such stopping points, producing what was called the inherent structure landscape.111,112
The stochastic time evolution represents the time evolution of an ensemble of random walkers as a Markov process, starting from some initial probability distribution over the states of the system; a special case would be to concentrate all the probability initially into one microstate.
In the example shown in the figure, the equilibration tree exhibits multiple sub-branchings even suggesting a self-similar structure, while the lid-based tree consisted of a simple main trunk to which all side-pockets were directly connected without any notable sub-branches.43
This approach is somewhat reminiscent of zone melting in the production of, e.g., high-quality low-defect semiconductor materials.
In fact, even modeling the dynamics in the presence of only energetic barriers assumes multiple attempts to cross the barrier in a quasi-equilibrium situation, leading to an Arrhenius-law type behavior.
Since in the ESA such searches are employed to study equilibration as a function of temperature within a pocket that serves as candidate of a locally ergodic region, this is no problem in that application. Also, the barriers we investigate would mostly be those that separate locally ergodic regions.
This procedure shares some features with the so-called deluge-algorithm113 where the random walker starts at very high energies and encounters a steadily decreasing lid until the walker ends up in some low-lying minimum. By increasing the lid value once no deeper minimum is found, the threshold algorithm avoids the trapping caused by the monotonically decreasing lid of the deluge-algorithm.
Picking a moveclass for a (stochastic) global optimization method is often considered the “black art” of global optimization.
Of course, this special design of moves is often guided by heuristic information about the system under investigation, and there exists the quite real danger of not reaching all relevant parts of the landscape.
