Transition Events in Butane Simulations: Similarities Across Models

From a variety of long simulations of all-atom butane using both stochastic and fully-solved molecular dynamics, we have uncovered striking generic behavior which also occurs in one-dimensional systems. We find an apparently universal distribution of transition event durations, as well as a characteristic speed profile along the reaction coordinate. An approximate analytic distribution of event durations, derived from a one-dimensional model, correctly predicts the asymptotic behavior of the universal distribution for both short and long durations.Comment: 18 pages, 6 figure

Rapid Determination of Multiple Reaction Pathways in Molecular Systems: The Soft-Ratcheting Algorithm

We discuss the ``soft-ratcheting'' algorithm which generates targeted stochastic trajectories in molecular systems with scores corresponding to their probabilities. The procedure, which requires no initial pathway guess, is capable of rapidly determining multiple pathways between known states. Monotonic progress toward the target state is not required. The soft-ratcheting algorithm is applied to an all-atom model of alanine dipeptide, whose unbiased trajectories are assumed to follow overdamped Langevin dynamics. All possible pathways on the two-dimensional dihedral surface are determined. The associated probability scores, though not optimally distributed at present, may provide a mechanism for estimating reaction rates

Theory of Systematic Computational Error in Free Energy Differences

Systematic inaccuracy is inherent in any computational estimate of a non-linear average, due to the availability of only a finite number of data values, N. Free energy differences (DF) between two states or systems are critically important examples of such averages in physical, chemical and biological settings. Previous work has demonstrated, empirically, that the ``finite-sampling error'' can be very large -- many times kT -- in DF estimates for simple molecular systems. Here, we present a theoretical description of the inaccuracy, including the exact solution of a sample problem, the precise asymptotic behavior in terms of 1/N for large N, the identification of universal law, and numerical illustrations. The theory relies on corrections to the central and other limit theorems, and thus a role is played by stable (Levy) probability distributions.Comment: 5 pages, 4 figure

Mediation in the Law Curriculum

Cited by Lord Neuberger in ‘Educating Future Mediators’ at the 4th Civil Mediation Council National Conference, May 201

Efficient Dynamic Importance Sampling of Rare Events in One Dimension

Exploiting stochastic path integral theory, we obtain \emph{by simulation} substantial gains in efficiency for the computation of reaction rates in one-dimensional, bistable, overdamped stochastic systems. Using a well-defined measure of efficiency, we compare implementations of ``Dynamic Importance Sampling'' (DIMS) methods to unbiased simulation. The best DIMS algorithms are shown to increase efficiency by factors of approximately 20 for a $5 k_B T$ barrier height and 300 for $9 k_B T$, compared to unbiased simulation. The gains result from close emulation of natural (unbiased), instanton-like crossing events with artificially decreased waiting times between events that are corrected for in rate calculations. The artificial crossing events are generated using the closed-form solution to the most probable crossing event described by the Onsager-Machlup action. While the best biasing methods require the second derivative of the potential (resulting from the ``Jacobian'' term in the action, which is discussed at length), algorithms employing solely the first derivative do nearly as well. We discuss the importance of one-dimensional models to larger systems, and suggest extensions to higher-dimensional systems.Comment: version to be published in Phys. Rev.