Energy Landscapes and Rare Events

Many problems arising from structural biology, theoretical chemistry, and condensed matter physics involve random walks on complex energy landscapes. The system spends most of time in metastable states, with thermally activated transitions from one metastable state to another. These transitions happen on a time scale specified by the Arrhenius law, which is much longer than the intrinsic time scale of the dynamical system. The questions of interest are the mechanism of such transition and the transition rates. In the past few years, we (Weinan E, Eric Vanden-Eijnden and myself) developed a theoretical framework and numerical methods (the zero- and finite-temperature string method) for dealing with such rare events in complex systems.

Smooth energy landscapes and the zero-temperature string method.   For systems with simple energy landscape in which the metastable states are separated by a few isolated barriers, the key objects are the transition states, which are saddle points on the potential energy landscape that separate the metastable states. These saddle points act as bottlenecks in a particular transition. The relevant notion for the transition pathways is that of minimum energy paths (MEPs). MEPs are paths in the configuration space that connects the metastable states along which the potential force is parallel to the tangent vector. The MEP allows us to identify the relevant saddle points, as well as the unstable directions at these points which are needed for the calculation of the prefactor in the transition rates. The zero-temperature string method (ZTS) is an effective tool for computing MEPs. Starting with an initial string, which is a curve in the configuration space connecting two metastable states, the method finds the MEP by repeating a two-step procedure (steepest descent and re-parametrization) until convergence. See the following references for more details.

Fig 1. An example of the smooth energy landscape and the minimum energy path.

• Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, W. E, W. Ren and E. Vanden-Eijnden, J. Chem. Phys. 126, 164103 (2007) 
• Higher order numerical scheme in the string method for finding minimum energy paths and saddle points, W. Ren, Commun. Math. Sci. 1, 377 (2003) 
• String method for the study of rare events, W. E, W Ren, and E. Vanden-Eijnden, Phys. Rev. B 66, 052301 (2002) 

Rough energy landscapes and the finite-temperature string method.   The problem of identifying transition pathways becomes much more challenging for systems with rough energy landscapes, as are the cases for typical chemical reactions of solvated systems and conformational changes of macro-molecules. In these situations, the traditional notion of transition states becomes inappropriate since the potential energy landscape typically contains numerous saddle points, most of which are separated by barriers that are less than or comparable to k BT, and therefore do not act as barriers. There may not exist specific microscopic configurations that identify the bottleneck of the transition. The most probable transition path is not unique; instead, a collection of paths are important. The finite-temperature string method (FTS) aims at solving problems of this type. The central objects here are the transition tube and the transition state ensemble , which are generalizations of the concepts of the MEP and the transition state. The FTS is designed to effectively solve the backward Kolmogorov equation for the committor function in the high-dimensional configuration space. Specifically, it determines a tube (the transition tube) by which transitions occur with high probability, and a family of hyperplanes within the tube which are approximations to the isocommittor surfaces. These are done by evolving a smooth curve using an averaged potential force in the configuration space. This curve converges to the center of the tube; the hyperplanes perpendicular to the converged curve are approximations to the isocommittor surfaces. The isocommittor surface with the value 1/2 and the Gibbs density function restricted to this surface define the transition state ensemble.

Fig 2. An example of the rough energy landscape (upper) and the transition tube (lower) identified by the string method.

• Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide, W. Ren, E. Vanden-Eijnden, P. Maragakis, and W. E, J. Chem. Phys. 123, 134109 (2005) 
• Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition Tubes W. E, W. Ren, and E. Vanden-Eijnden, Chem. Phys. Lett. 413, 242 (2005) 
• Finite temperature string method for the study of rare events, W. E, W. Ren, and E. Vanden-Eijnden, J. Phys. Chem. B 109, 6688 (2005) 

Transition pathways in non-gradient systems and finite-time switching.   The Wentzell-Freidlin theory of large deviations provides a theoretical foundation for identifying the transition pathways associated with a dynamical system perturbed by a small noise. In particular, it gives an estimate on the probability of the transition paths in terms of an action functional. This estimate shows that the most probable transition pathway can be obtained by the (constrained) minimization of the action. This path is called the minimum action path (MAP). The minimum action method is an effective tool for computing the MAP for a given switching time. The MAP reduces to the MEP for gradient systems and infinite switching time.

• Minimum action method for the Kardar-Parisi-Zhang equation, H. C. Fogedby and W. Ren, Phys. Rev. E 80, 041116 (2009)  
• Adaptive minimum action method for the study of rare events, X. Zhou, W. Ren and W. E, J. Chem. Phys. 128, 104111 (2008)  
• Minimal action method for the study of rare events, W. E, W. Ren and E. Vanden-Eijnden, Commun. Pure Appl. Math. 57, 637 (2004) 

Some applications of the string method.   The string method has been applied to a variety of problems arising from different disciplines. These include the study of the energy landscape of ferromagnetic thin films, conformational changes of biomolecules, current dissipation in thin superconducting wires, dislocation motion in crystalline solids, and quantum metastability induced by tunneling.

• Computing transition rates of thermally activated events in dislocation dynamics, C. Jin, W. Ren and Y. Xiang, Scripta Materialia, 62, 206 (2010)
• Application of the string method to the study of critical nulei in capillary condensation, C. Qiu, T. Qian and W. Ren, J. Chem. Phys., 129, 154711 (2008)  
• Phase slips in superconducting wires with nonuniform cross section: A numerical evaluation using the string method, C. Qiu, T. Qian and W. Ren, Phys. Rev. B, 77, 104516 (2008)  
• Numerical study of metastability due to tunneling: The quantum string method, T. Qian, W. Ren, J. Shi, W. E and P. Sheng, Physica A, 379, 491 (2007) 
• Current dissipation in thin superconducting wires: Accurate numerical evaluation using the string method, T. Qian, W. Ren and P. Sheng, Phys. Rev. B, 72, 014512 (2005) 
• Energy landscape and thermally activated switching of submicron-sized ferromagnetic elements, W. E, W. Ren and E. Vanden-Eijnden, J. Appl. Phys. 93, 2275 (2003)  
• Numerical study of dislocation dynamics using the string method, C. Jin, W. Ren and Y. Xiang, preprint