Stochastic path sampling

Further efficiency is obtained by making the path length variable [87]. As only shots from the barrier itself are useful in stochastic path sampling, it is natural to stop with the integration of the equation of motion once one reaches a stable state, provided that the trajectory is then really committed to a stable state [87,88]. 

Crooks, G. E. Chandler, D., Efficient transition path sampling for nonequilibrium stochastic dynamics, Phys. Rev. E 2001, 64, 026109/1 -4... 

The perspective exploited by transition path sampling, namely, a statistical description of pathways with endpoints located in certain phase-space regions, was hrst introduced by Pratt [27], who described stochastic pathways as chains of states, linked by appropriate transition probabilities. Others have explored similar ideas and have constructed ensembles of pathways using ad hoc probability functionals [28-35]. Pathways found by these methods are reactive, but they are not consistent with the true dynamics of the system, so that their utility for studying transition dynamics is limited. Trajectories in the transition path ensemble from Eq. (1.2), on the other hand, are true dynamical trajectories, free of any bias by unphysical forces or constraints. Indeed, transition path sampling selects reactive trajectories from the set of all trajectories produced by the system s intrinsic dynamics, rather than generating them according to an artificial bias. This important feature of the method allows the calculation of dynamical properties such as rate constants. 

The transition path sampling techniques we have described assume that an initial reactive pathway is available. Generating such a pathway is therefore an important step in applying the method. In the simplest cases, a trajectory connecting A and B can be obtained by running a long molecular dynamics (or stochastic dynamics) simulation. For most applications, however, the... 

Once a stochastic molecular dynamics-based framework is developed it is possible to modify it or use it as a building block in order to enhance sampling efficiency. One can for example incorporate these methods into kinetic Monte-Carlo, other metropolized schemes, or various advanced path sampling techniques. Thus the design of good molecular SDE methods is the cornerstone of much of modern molecular modelling. 

FIGURE 3.3.2 Film drainage under the action of a stochastic force. Sample paths of the process, one leading to aggregation and the other to separation. 

Here the mean value ( ) has to be taken over the possible random forces occurring in the sample society via the stochastic transitions of its members. The stochastic nature of the random forces on the rhs of (2.35) in turn leads to non-deterministic, stochastic paths of the variable x(t) Paths x(t) develop differently under the influence of the stochastic forces (t) in (2.35) even if they start out from the same initial value jk (0). The mean deviation and mean square deviation of x (t) from the initial value after a short time interval At can, however, be calculated by integrating (2.35) iteratively over with the initial... 

In the simple sampling procedure of generating chain conformations all successfully generated walks have equal probabihty. Walks are grown purely stochastically. Each time an attempted new bond hits a site which is already occupied, one has to start at the very beginning. Otherwise different conformations would have different probabihties and this would introduce an effective attraction among the monomers [54]. With this method, each conformation is taken randomly out of the q q — 1) possible random paths which do not include direct back-folding. However, the total number of SAW on a lattice is known [26] to be ... 

We will first follow the decay paths taken during several individual runs, just to see how they can vary. Then we will examine the behavior of larger samples to find actual values for cpf and cellular automata models are stochastic, the results for and small samples will likely differ significantly from the deterministic values cited above. The differences between the observed and the deterministic values will normally decrease as the sample size is increased. We will also examine the observed lifetimes Xf and tp of the decays of the Si and T i states (Chapter 7) and compare the values found with the corresponding deterministic values. 

A number of approaches have been used to try to improve the convergence of the JE in cases where it is problematic. Some of these are analogues of processes introduced to improve sampling in the FEP approaches. For example, Adjanor et al introduce path-biasing schemes to improve convergence. They consider a stochastic system, and carry out tests with simulations on clusters of LJ particles. 

Various deterministic and stochastic sampling techniques for path ensembles have been proposed [4-6]. Here we consider only Monte Carlo methods. It is important, however, to be aware that while the path ensemble is sampled with a Monte Carlo procedure each single pathway is a fully dynamical trajectory such as one generated by molecular dynamics. 

In (25) it is required that G contains Do among its members. The has the same meaning as before all vertices in G must be visited by the paths. Using (24) and (19) leads to an expression for Eq suitable for a stochastic sampling of graphs, analogous to (15) ... 

Zuckerman, D.M., Woolf, T.B. Dynamic reaction paths and rates through importance-sampled stochastic dynamics, J. Chem. Phys. 1999,111(21), 9475-84. 

It is possible to sample stochastic transition pathways by making only local displacements of trajectories. For example, a randomly chosen time shce of an existing pathway may be modified by adding a small displacement 5x to positions and momenta, xj" = xj" + x. All other time slices remain unchanged. This modification, which is local in time, gives a different but finite path probability If the displacement x is chosen from a... 

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