System point trajectories

However, not all space curves singly connecting reactant and product asymptotes correspond to a realistic time evolution describing an elementary process. Such evolution is determined by the equations of motion within the quasiclassical approach the space curves can be interpreted as system point trajectories in M with their end points located in the reactant and product asymptotes 43,44), a trajectory Q — Q(t) is then determined by the classical equations of motion [i.e., within some of the equivalent formulations of classical mechanics tantamount to Eq. (9)]. 

The way in which reactant change into products in an elementary process will be regarded as the microscopic collision mechanism of the elementary process in question. It is determined within the quasiclassical approach by the characteristics of the system point trajectories. 

If the end points are defined, the solution to the classical equations of motion corresponding to an elementary process can be sought the resulting system point trajectories represent the realistic evolution of the polyatomic system from reactants to products on the given potential energy surface Wm(QJ) at the total energy H — T + W(Q ). The solutions to the equations of motion can thus be considered as transformations leading from a set of boundary conditions in the reactant asymptote to a set of boundary conditions in the product asymptote. 

Whenever a seam is encountered, the system point trajectory Q = Q(l) is split into n branches Qe, each branch evolving under the influence of one of the interacting potential energy surfaces. A system point trajectory connecting the reactant and product asymptotes may arrive at several seams and its splitting may occur at each seam. The electronic transition probabilities Pt(Q, Q) can be calculated by numerical integration of the relations, Eq. (14), at each seam... 

Consequently, the electronic transition probabilities can be interpreted as relative weights of the individual branches of a system point trajectory. The final weights of trajectories terminating in the product asymptote can then serve for calculating the reaction attributes in the same manner as the numbers of trajectories in case of adiabatic processes (7,49). 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Pagels show that thermodynamic depth is proportional to the difference between the state s thermodynamic entropy (i.e. its coarse grained entropy) and its finegrained entropy, given by fcex volume of points in phase space corresponding to the system s trajectory, where k], is Boltzman s constant. 

To proceed, we begin with the fully quantum mechanical description of the system in terms of a Keldysh action for a single variable [10, 9], that incorporates information about FCS of the normal conductor and the properties of the Josephson junction. We calculate the escape rate by considering saddle-point trajectories of the action, A, that connect the potential minimum with the nearest potential maximum. With exponential accuracy, the rate is given by T exp(— ImA/ft). 

In the case of steady state bifurcations, certain eigenvalues of the linear-approximation matrix reduce to zero. If we consider relaxations towards a steady state, then near the bifurcation point their rates are slower. This holds for the linear approximation in the near neighbourhood of the steady state. Similar considerations are also valid for limit cycles. But is it correct to consider the relaxation of non-linear systems in terms of the linear approximations To be more precise, it is necessary to ask a question as to whether this consideration is sufficient to get to the point. Unfortunately, it is not since local problems (and it is these problems that can be solved in terms of the linear approximations) are more simple than global problems and, in real systems, the trajectories of interest are not always localized in the close neighbourhood of their attractors. 

Fig. 5.1.1 Configuration space showing the regions for reactants (r) and products (pj). ti is an example of a trajectory that leads to the formation of products pi. S(pl,r) is part of the dividing surface between r and pi, where the system points have an outward velocity from the reactant region.

We consider an ensemble of systems each containing n atoms. Thus, q = (<71, , < 3n), P = (pi, , P3n), and dpdq = Ilf" (dp dqi). We assume that all interactions are known. As time evolves, each point will trace out a trajectory that will be independent of the trajectories of the other systems, since they represent isolated systems with no coupling between them. Since the Hamilton equations of motion, Eq. (4.63), determine the trajectory of each system point in phase space, they must also determine the density p(p, q, t) at any time t if the dependence of p on p and q is known at some initial time to. This trajectory is given by the Liouville equation of motion that is derived below. 

A gradient path of V has a simple physical interpretation. It is a line of force—the path traversed by a test charge moving under the influence of the potential F(r X). At a critical point other than a (3, — 3) critical point, the force vanishes. Thus a critical point in the field V(r X) denotes a point of electrostatic balance between the attractors of the system. Since trajectories defining the surface which separates neighbouring basins satisfy the zero-flux condition... 

Ensemble of System Points Moving Along Their Trajectories in Phase Space... 

The analysis of linearized sytem thus allows, when conditions (1)—(3) are met, us to find the shape of phase trajectories in the vicinity of stationary (singular) points. A further, more thorough examination must answer the question what happens to trajectories escaping from the neighbourhood of an unstable stationary point (unstable node, saddle, unstable focus). In a case of non-linear systems such trajectories do not have to escape to infinity. The behaviour of trajectories nearby an unstable stationary point will be examined in further subchapters using the catastrophe theory methods. 

As we have concluded in Chapter 5, in the case of the Lorenz system a trajectory always remains within a confined region of the phase space, being non-periodical. For t - oo, the trajectory approaches a certain limit set the Lorenz attractor. It follows from the Liouville theorem that the Lorenz attractor has a zero volume (since divF < 0). This implies that, apparently, the Lorenz attractor is a point (dimension zero), a line (dimension one), or a plane (dimension two). Then, however, the trajectory for t -> oo would have remained within a confined region on the plane and, by virtue of the Poincare-Bendixon theorem. Hence, a conclusion follows that the Lorenz attractor has a fractional dimension, larger than two. 

Figure 2.3 shows the zeroth-, first-, and second-order new actions along the trajectory at O.OSe. At even E = 0.05e, only slightly above the saddle point energy, almost of all the zeroth-order actions J°" (p,q) do not maintain constancy of motion at all. This result implies that the system s trajectory reflects even very small nonlinearities on the PES and deviates from a simple normal mode picture. As we extend the order of LCPT, some but not all LCPT actions tend to be conserved in the saddle region. 

In general, the operations optimization task is to find suitable set point trajectories for the controllers. As the controllers are omitted from our simplified TMP system model, no setpoint optimization is included in the study. However, the refiner scheduling optimization can also be considered as operations optimization with refiner activity set point trajectory as binary valued (on/off) function of time. 

In the control law part, the MFC eontroller has to calculate the set of control moves (Am) into the future that allows the system to follow a predefined set-point trajectory (from off-line optimization). This is done by solving for the quadratic cost function... 

The solution of the initial value problem described by Eq. (2.9) can be visualised so that the calculated concentrations are plotted as a function of time as shown in Fig. 2.1a. Another possibility is to explore the solution in the space of concentra-tirais as in Fig. 2.1b. In this case, the axes are the concentrations and the time dependence is not indicated. The actual concentration set is a point in the space of concentrations. The movement of this point during the simulation outlines a curve in the space of cOTicentrations, which is called the trajectory of the solution. This type of visuaUsation is often referred to as visualisation in phase space. In a closed system, the trajectory starts from the point that corresponds to the initial value and after a long time ends up at the equilibrium point. In an open system where the reactants are continuously fed into the system and the products are continuously removed, the trajectory may end up at a stationary point, approach a closed curve (a limit cycle in an oscillating system) or follow a strange attractor in a chaotic system. It is not the purpose of this book to discuss dynamical systems analysis of chemical models in detail, and the reader is referred to the book of Scott for an excellent treatment of this topic (Scott 1990). 

At this point, the system behavior trajectory departs from the equilibrium liquid line, causing the "glass transition". With a slower cooling rate, the periods for the reestablishment of equilibrium between temperature jumps are longer, and the equilibrium line is followed to lower temperatures before glassy response is observed. In slower experiments, then, the model predicts that the glass transition occurs at lower temperatures due to the interplay of the time scale of the experiment and the kinetics of recovery. 

The system s trajectory follows a path below the ii = —Q line (i.e., reversed rotation of the lead screw) until it reaches the N = 0 line again and the lead screw rotation is once again seized. This cycle continues until the point —T R/kmr tanl + cQ./k), —Q) is reached, where the N = 0 line intersects the horizontal u = —Q line. Also note that initial motion from conditions where T R + mr tan l ku 0) + cm(0)) < 0 i.e. N <0) and m(0) < — Q i.e. 6 <0) is also not possible and the system s trajectory transfers instantaneously to (m(0), —Q) from which the motion is governed by (8.22) and continues towards origin with negative N and ii t) + Q > 0. 

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