Classical Trajectories with Boundary Value Formulation Elber

Ron Elber Calculation of Classical Trajectories with Boundary Value Formulation, Lect. Notes Phys. 703, 435-451 (2006) 

An algorithm to compute classical trajectories using boundary value formulation is presented and discussed. It is based on an optimization of a functional of the complete trajectory. This functional can be the usual classical action, and is approximated by discrete and sequential sets of coordinates. In contrast to initial value formulation, the pre-specified end points of the trajectories are useful for computing rare trajectories. Each of the boundary-value trajectories ends at desired products. A difficulty in applying boundary value formulation is the high computational cost of optimizing the whole trajectory in contrast to the calculation of one temporal frame at a time in initial value formulation. 

Calculation of classical trajectories of molecular systems is a powerful tool for stud3dng the thermod3mamics and the kinetics of biophysical problems. Indeed the availability of numerous computer packages and journals devoted to such simulations suggests that these studies are extremely popular, and are influencing many fields. It is therefore no surprise that a number of chapters in this book are devoted to the use of classical d3mamics in condensed matter physics. 

The emphasis in other chapters of this book is on initial value solution of classical equations of motion (e.g. the Newton s equations). The Newton s equations are second order differential equations - MX = —dU/dX, where X [X e is the coordinate vector. Throughout this chapter X is assumed to be a Cartesian vector, M is a 3N x 3N (diagonal) mass matrix, and U is the potential energy. A widely used algorithm that employs the coordinates and the velocities (V) at a specific time and integrates the equations of motion in small time steps is the Verlet algorithm [1]  

The small size of the time step restricts the calculation to rapid processes. Even for fast molecular events (nanoseconds) millions of femtosecond steps are required to reach the nanosecond time scale. Longer time scales are important for simulating many biophysical events and are not reachable with routine calculations (e.g. rapid protein folding occurs at the microseconds and milliseconds time scales). 

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