Optimal trajectory

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 improve the accuracy of the solution, the size of the time step may be decreased. The smaller is the time step, the smaller are the assumed errors in the trajectory. Hence, in contrast (for example) to the Langevin equation that includes the friction as a phenomenological parameter, we have here a systematic way of approaching a microscopic solution. Nevertheless, some problems remain. For a very large time step, it is not clear how relevant is the optimal trajectory to the reality, since the path variance also becomes large. Further-... 

We shift the discussion from computations of optimal trajectories to sampling of trajectories with a given weight. Hence rather than seeking a single trajectory with a maximal, we consider in addition other trajectories with... 

We proposed [7] two possible approaches to estimate these errors. Here we discuss them only briefly. Trajectories that are not too far from the optimal trajectory will have a significant weight. We denote by Xopt t) the optimized trajectory, and by Xexact t) the exact trajectory. The optimal trajectory is not the same as the exact trajectory, since it was computed with a large time step. SjlcP is expanded up to a second order near the optimal trajectory... 

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. 

It is also possible to use normal mode analysis [7] to estimate the difference between the exact and the optimal trajectories. Yet another formula is based on the difference between the optimal and the exact actions 2a w [5[Yeract(t)] (f)]]- The action is computed (of course), employ-... 

Another way of visualizing the optimal trajectory for the exothermic reversible reaction is to consider isothermal reaction rates at increasing temperatures. At T the equilibrium conversion is high but the rate is low, at T2 the rate is higher but the equihbrium conversion is lower, at the rate has increased further and the equihbrium conversion is even lower, and at a high temperature T4 the initial rate is very high but the equihbririm conversion is very low. 

The reaction coordinate ----, a more optimal trajectory for tunneling. The arrows show the... 

In the presence of weak noise there is a finite probability of noise-induced transitions between the chaotic attractor and the stable limit cycle. In Fig. 14 the filled circles show the intersections of one of the real escape trajectories with the given Poincare section. The following intuitive escape scenario can be expected in the Hamiltonian formalism. Let us consider first the escape of the system from the basin of attraction of a stable limit cycle that is bounded by an saddle cycle. In general, escape occurs along a single optimal trajectory qovt(t) connecting the two limit cycles. 

If the noise is weak, then the probability P exp —S/D) to escape along the optimal trajectory is exponentially small, but it is exponentially greater than the escape probability along any other trajectory, including along other heteroclinic trajectories of the system (37). 

Since the basin of attraction of the CA is bounded by the saddle cycle SI, the situation near SI remains qualitatively the same and the escape trajectory remains unique in this region. However, the situation is different near the chaotic attractor. In this region it is virtually impossible to analyze the Hamiltonian flux of the auxiliary system (37), and no predictions have been made about the character of the distribution of the optimal trajectories near the CA. The simplest scenario is that an optimal trajectory approaching (in reversed time) the boundary of a chaotic attractor is smeared into a cometary tail and is lost, merging with the boundary of the attractor. 

It is evident that all real trajectories pass through the close neighborhood of some optimal trajectory in a tube with a radius cx /7). Therefore it is possible to determine the optimal escape paths by simple averaging performed separately for each group of trajectories. The number of different optimal escape paths obtained for the transition CA S3 depends on the choice of the working point. From one to three distinct optimal escape paths for operation in various regimes were observed experimentally. The escape probabilities along different paths are different, and, as the noise intensity is reduced, one of the escape paths becomes exponentially more probable then the others. In what follows we concentrate on the properties of this most probable escape path. 

E.E.Nikitin and M.Ya.Ovchinnikova, Optimal trajectory approach in the theory of photodissociation of thermally excited molecules. Chem. Phys. 138,45 (1989)... 

E.E.Nikitin, Optimal trajectory approach in the theory of collisional vibrational relaxation of diatomic molecules, in Dynamical Processes in Moleailar Physics, First EPS Southern European School of Physics, ed. G. Delgado-Barrio, Bristol and Philadelphia, Institute of Physics Publishing Ltd, 1993, p. 55... 

The manipulated variables for the optimization are the mass flux over the membrane (J) and the ultrasound utilization rju). Figure 7 shows the optimal trajectories of the manipulated variables. The results illustrate the strength of the task based design approach. In this case a very tight constraint on the growth rate has been imposed. It can be seen how both the flux and the utilization of ultrasound work together to maximize the crystal mean size. In the initial phase the ultrasound and flux are both high to create... 

At this point we may continue in one of two directions. We may use a single approximate trajectory at the neighborhood of the exact trajectory that is, the trajectory that was obtained by the minimization of the discrete action. Alternatively, we recognize that the exact trajectory deviates from the optimal trajectory by errors distributed normally (keep in mind that the error distribution is a property of the exact trajectory). We may sample errors (and plausible trajectories) from the appropriate distribution of coordinates in the neighborhood of the trajectory with filtered high-frequency modes. The sampling in the neighborhood of the optimized trajectory should be normalized (we approximate one trajectory) ... 

Maximization of final product quality and minimization of drying time. e constraint method with SQP Optimal trajectories of air temperature and relatiye humidity for drying paddy rice were determined. Olmos el al. (2002)... 

Chang, C. S. (1966), Discrete-Sample Curve Fitting Using Chebyshev Polynomials and the Approximate Determination of Optimal Trajectories via Dynamic Programming, IEEE Transactions on Automatic Control, Vol. AC-11, pp. 116-118. 

The detomination of trajectories for the addition of monomers and/or initiators, and/or control of temperature in batch or semi-batch polymerizations are almost alwa rs dme off-line. These trajectories may be the result of operating experience, or they may be developed by calculating optimal trajectories to achieve certain goals (reduced kettle time, desired MMD or copolymer composition distribution (CCD)) subject to constraints on heat transfer capacity, etc. [33]. The rigorous calculation of optimal trajectories requires a reasonably accurate model of the polymerization process. 

The optimal trajectory is that of a free particle, i.e., the straight line connecting the points X and y e S2q, where is the support of the initial condition ... 

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