Configuration space trajectories

Figure 9.17 The probability densities, ip 2, in the top row are found to resemble the configuration space trajectories in the bottom row. The states are from the JV = 5 polyad of H2O and are numbered in increasing energy order from 1 to 6 in correspondence with the numbered trajectories on the polyad phase sphere (from Xiao and Kellman, 1989).

In a Brownian dynamics machine simulation, the configuration-space trajectories are composed of successive displacements rf —> rj over a short time-step At. The equation of motion is ... 

Fig. 5. Schematic illustration of a configuration space trajectory, t, for a chemical process AB + C A + BC. The solid lines depict...

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

But a computer simulation is more than a few clever data structures. We need algorithms to manipulate our system. In some way, we have to invent ways to let the big computer in our hands do things with the model that is useful for our needs. There are a number of ways for such a time evolution of the system the most prominent is the Monte Carlo procedure that follows an appropriate random path through configuration space in order to investigate equilibrium properties. Then there is molecular dynamics, which follows classical mechanical trajectories. There is a variety of dissipative dynamical methods, such as Brownian dynamics. All these techniques operate on the fundamental degrees of freedom of what we define to be our model. This is the common feature of computer simulations as opposed to other numerical approaches. 

In the transition path sampling method we are interested in trajectories that start in a certain region of configuration space, which we will call region si, and end in another region, 38. We call such trajectories reactive. Accordingly, we restrict the probability distribution from (7.3) to reactive trajectories only (see Fig. 7.2)... 

As alluded to above, the method relies upon the identification of a 3A-dimensional vector in configuration space, es, which points along a favored direction for the motion of the system. We then choose the initial momenta for the trajectory ensemble from a Gaussian distribution artificially extended in the direction of es, as illustrated in the right panel of Fig. 8.2. In the case of free energy reconstructions from (8.49), we wish to induce motion along a predefined pulling direction, and so es can be found by inspection. 

Overall sampling quality — quantitative analysis. For dynamical trajectories, the "structural decorrelation time" analysis [10] can estimate the slowest timescale affecting significant configuration-space populations and hence yield the effective sample size. For non-dynamical simulations, a variance analysis based on multiple runs is called for [1]. Analyzing the variance in populations of approximate physical states appears to be promising as a benchmark metric. 

Let us consider a stochastic system described by a generic variable C. This variable may stand for the position of a bead in an optical trap, the velocity field of a fluid, the current passing through a resistance, of the number of native contacts in a protein. A trajectory or path V in configurational space is described by a discrete sequence of configurations in phase space. 

It is hard to believe that, in order to see how the enzyme works, or how the protein folds up, one must view the movie in its entirety. It is more plausible that there are only a few interesting parts, during which the system passes through critical bottlenecks in its configuration space the rest of the time being spent exploring large, equilibrated reservoirs between the bottlenecks. If the trajectory calculation were... 

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