Discrete trajectory

Figure C3.6.2 (a) The (fi2,cf) Poincare surface of a section of the phase flow, taken at ej = 8.5 with cq < 0, for the WR chaotic attractor at k = 0.072. (b) The next-amplitude map constmcted from pairs of intersection coordinates. ..,(c2(n-l-l),C2(n-l-2),C2(n-l-l)),...j. The sequence of horizontal and vertical line segments, each touching the diagonal B and the map, comprise a discrete trajectory. The direction on the first four segments is indicated.

The Crooks relation follows from an elegant derivation of Jarzynski s identity using path-sampling ideas [18], For instructive purposes, that derivation is briefly summa-rized here. Consider generating a discrete trajectory z0 - z i . .. z v, where... 

This sequence of states is a discrete representation of the continuous dynamical trajectory starting from zo at time t = 0 and ending at z at time t = . Such a discrete trajectory may, for instance, result from a molecular dynamics simulation, in which the equations of motion of the system are integrated in small time steps. A trajectory can also be viewed as a high-dimensional object whose description includes time as an additional variable. Accordingly, the discrete states on a trajectory are also called time slices. 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

In this chapter, we show that a symplectic integrator can be viewed as being effectively equivalent to the flow map of a certain Hamiltonian system. The starting point is that symplectic integrators are symplectic maps that are near to the identity since they depend on a parameter (the stepsize h) which can be chosen as small as needed, and, if consistent, in the limit -> 0, such a map must tend to the identity map. We can express the fundamental consequence as follows not only are Hamiltonian flow maps symplectic, but also near-identity symplectic maps are (in an approximate sense) Hamiltonian flow maps [31], The fact leads to the existence of a modifled (perturbed) Hamiltonian from which the discrete trajectory may be derived (as snapshots of continuous trajectories). In some cases we may derive this perturbed Hamiltonian as an expansion in powers of the stepsize. 

Actually, as explained previously, we must interpret this carefully a finite truncation of this expansion may characterize the discrete trajectories observed in practice, but the series expansion itself does not typically converge. Assuming such a perturbative expansion is valid, the issue of ergodicity of the numerical method cm be related to the corresponding issue for the dynamical system characterized by H. ... 

The dynamics are discretized using a standard Langevin dynamics second-order algorithm taken from [62], where we sample the distribution of q along our discretized trajectory. Each step of the algorithm requires one force evaluation, so we may think of the vertical axis in Fig. 7.2 representing the total computational work. 

A numerical technique that has become very popular in the control field for optimization of dynamic problems is the IDP (iterative dynamic programming) technique. For application of the IDP procedure, the dynamic trajectory is divided first into NS piecewise constant discrete trajectories. Then, the Bellman s theory of dynamic programming [175] is used to divide the optimization problem into NS smaller optimization problems, which are solved iteratively backwards from the desired target values to the initial conditions. Both SQP and RSA can be used for optimization of the NS smaller optimization problems. IDP has been used for computation of optimum solutions in different problems for different purposes. For example, it was used to minimize energy consumption and byproduct formation in poly(ethylene terephthalate) processes [ 176]. It was also used to develop optimum feed rate policies for the simultaneous control of copolymer composition and MWDs in emulsion reactions [36, 37]. 

This iterative procedure assumes that the electronic structure is recomputed and the full diagonalization of the Hamiltonian is performed at each time step t along the discrete trajectory R/(f). ... 

H. H. Tan and R. B. Potts, A Discrete Trajectory Planner for Robotic Arms with Six Degrees of Freedom, IEEE Transection on Robotics and Automation, Vol. 5. No. 5. October 1989. pp. 681-690. 

Figure Cl.5.8. Spectral jumping of a single molecule of terrylene in polyethylene at 1.5 K. The upper trace displays fluorescence excitation spectra of tire same single molecule taken over two different 20 s time intervals, showing tire same molecule absorbing at two distinctly different frequencies. The lower panel plots tire peak frequency in tire fluorescence excitation spectmm as a function of time over a 40 min trajectory. The molecule undergoes discrete jumps among four (briefly five) different resonant frequencies during tliis time period. Arrows represent scans during which tire molecule had jumped entirely outside tire 10 GHz scan window. Adapted from...

We now examine how a next-amplitude-map was obtained from tire attractor shown in figure C3.6.4(a) [171. Consider tire plane in tliis space whose projection is tire dashed curve i.e. a plane ortliogonal to tire (X (tj + t)) plane. Then, for tire /ctli intersection of tire (continuous) trajectory witli tliis plane, tliere will be a data point X (ti + r), X (ti + 2r))on tire attractor tliat lies closest to tire intersection of tire continuous trajectory. A second discretization produces tire set Xt- = k = 1,2,., I This set is used in tire constmction... 

Fig. 1. Comparison of two different dynamical simulations for the Butane molecule Verlet discretization with stepsize r = O.OOSfs. Initial spatial deviation 10 A. Left Evolutions of the total length (=distance between the first and the last carbon atom) of the molecule (in A). Right Spatial deviation (in A) of the two trajectories versus time.

Summarizing, from a mathematical point of view, both forward and backward analysis lead to the insight that long term trajectory simulation should be avoided even with symplectic discretizations. Rather, in the spirit of multiple as opposed to single shooting (cf. Bulirsch [4, 18]), only short term trajectories should be used to obtain reliable information. 

Hence, we use the trajectory that was obtained by numerical means to estimate the accuracy of the solution. Of course, the smaller the time step is, the smaller is the variance, and the probability distribution of errors becomes narrower and concentrates around zero. Note also that the Jacobian of transformation from e to must be such that log[J] is independent of X at the limit of e — 0. Similarly to the discussion on the Brownian particle we consider the Ito Calculus [10-12] by a specific choice of the discrete time... 

A saddle point approximation to the above integral provides the definition for optimal trajectories. The computations of most probable trajectories were discussed at length [1]. We consider the optimization of a discrete version of the action. 

To compute the above expression, short molecular dynamics runs (with a small time step) are calculated and serve as exact trajectories. Using the exact trajectory as an initial guess for path optimization (with a large time step) we optimize a discrete Onsager-Machlup path. The variation of the action with respect to the optimal trajectory is computed and used in the above formula. 

A single calculation of the discrete path integral with a fixed length of time t can be employed to compute the state conditional probability at many other times. It is possible to use segments of the path of time length At, 2At,..., NAt sampled in trajectories of total length of NAt and to compute the corresponding state conditional probabilities. The result of the calculations will make it possible to explore the exponential relaxation of P Ao B,t) for times between 0 and t. 

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

Tn general, the. solvent-accessible surface (SAS) represents a specific class of surfaces, including the Connolly surface. Specifically, the SAS stands for a quite discrete model of a surface, which is based on the work of Lee and Richards [182. They were interested in the interactions between protein and solvent molecules that determine the hydrophobicity and the folding of the proteins. In order to obtain the surface of the molecule, which the solvent can access, a probe sphere rolls over the van der Waals surface (equivalent to the Connolly surface). The trace of the center of the probe sphere determines the solvent-accessible surjace, often called the accessible swface or the Lee and Richards surface (Figure 2-120). Simultaneously, the trajectory generated between the probe and the van der Waals surface is defined as the molecular or Connolly surface. 

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in-... 

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