This week we will take a look at a modern interpretation of the study of rare event pathways. Previously we saw that transitions representing rare events can be defined in terms of a path across the free energy plane of the system. Wigner's method is only useful with small systems and correspondingly small degrees of freedom. Weinan E, Weiqing Ren, and Eric Vanden-Eijnden proposed a method for expanding this theory into larger systems with many degrees of freedom in their 2002 paper
The string method begins by connecting two metastable states of interest with a line in configuration space. The configuration space is the minimum subset of the free energy space which still describes the transition (finding the coordinates for large systems is still an active area of research). The initial string is then iteratively evolved, at n discrete points, according to the underlying physics of the system. At each iteration the arclength of the string is computed and the string is re-parameterized to enable efficient sampling of the configuration space. The computation is stopped once the total movement of the string is below a specified threshold.
By reducing the number of variables necessary to represent the transition the string method allows for efficient computation of the transition pathway. The paper presents two test cases which demostrate the effectiveness of the method. While these test systems are simple the show important facts about the method and allow for analytical solutions to be the basis of comparison.
One way to improve the paper would have been to show a contour plot of the energy landscape with the string overlaid. A detailed plot would provide a clear picture of the mechanics of the method.
"Characterization of a Dynamic String Method for the Construction of Transition Pathways in Molecular Reacions" [doi: 10.1021/jp212611k] was published on May 22, 2012 and builds on the string method. Hummer's paper looks interesting because it takes a dynamic approach to sampling the configuration space and updating the string.