Variational Principles in Conformation Dynamics - FU Berlin, FB MI

V. Summary and outlookWe have presented a method to approximate the eigenvalues and time scales of slow processesin stochastic dynamical systems based on the variational principle Theorem II.6 andthe Roothan-Hall method Theorem II.10. WehaveappliedthismethodwithGaussianfunctionsto one-dimensional and a few simple higher dimensional examples of diffusion processesgoverned by Smoluchowski dynamics. So far, we have seen that in all cases a convincingestimate of the involved time scales was achieved. Comparison with results obtained fromMarkov state model discretizations showed that the variational method can reproduce thequality of these.V.1. Future workFuture work on the subject will have to deal with a number of questions:V.1.1. Application to larger systemsCan we, in a chain with many internal coordinates, obtain a convincing result by simply addingup local basis functions, dependent on one coordinate, chosen on the basis of the molecularenergy function? This assumes that the internal coordinates are sufficiently independent toallow this approximation. If it works, it would avoid the use of clustering techniques andresult in a computationally affordable method to compute the dominant eigenvalues andrelevant time scales. As a first test, we will have to try a five atom system with additionalCoulomb interactions, and see if it works.