The dynamics of complex systems are often driven by rare but important events. Well-known examples include nucleation events during phase transitions, conformational changes in macromolecules, and chemical reactions. The long time scale associated with these rare events is a consequence of the disparity between the effective thermal energy and typical energy barrier of the systems. The dynamics proceeds by long waiting periods around metastable states followed by sudden jumps from one state to another.

The problem of finding saddle points on a multidimensional energy surface can be broadly divided into two categories - (a) systems for which we have an intuitive idea of the initial and final metastable configurations corresponding to a transition event and the path joining these contains at least one saddle point, and (b) systems for which we only have the initial state lying close to a locally stable fixed point on the energy surface. The later problem involves exploring the PES to search for low barrier saddle points and other locally stable configurations. In terms of algorithmic complexity the problem of determining the transition path and the saddle point is much simpler for category (a) than for problems in category (b).

In general, while dealing with complex systems one has very little information about the final states or the ensuing transition pathways and least of all the transition states. For example, while studying grain boundary migration or dislocation emission from grain boundaries it is very diffi- cult to have prior knowledge of the final state. On the other hand, it is often desirable to obtain a list of possible low energy barriers and their corresponding structure to have a pragmatic view of the probable mechanisms by which a physical process can take place. For example, there is considerable interest in understanding the atomistic mechanisms related to softening of the lat- tice during melting of a crystal. The quantitative role of point defects (interstitials, vacancies, etc.) and thermal fluctuations is still an active topic of research. Similar, situation is encoun- tered during nanoindentation. There exists considerable literature reports of deformation processes governed by dislocation activity during initial states of nanoindentation. However, the activa- tion volumes reported from nanoindentation experiments show higher strain-rate and temperature sensitivity. Hence to have a better understanding of the dominant deformation mechanisms it is imperative to have a list of plausible point defect activity. In some of these cases, one can resort to algorithms like hyperdynamics, bond-boost method and temperature accelerated dynam- ics (TAD), parallel replica method to obtain quantitative insights into the long time dynamical behavior of a system.

This web-page contains an algorithm to sample saddle configurations lying close a local minimum point on the energy surface.

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Send comments to Amit Samanta: asamanta AT math.princeton.edu or Weinan E: weinan AT math.princeton.edu

-Last updated on October 5, 2011.

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