Phase space plane

Figure 5. The distributions of the recrossing trajectories over configurational surface S qi = 0) at time t = 0 on the phase-space planes (pf (p,q), (p,q)) at E = 0.5e, where most modes are strongly chaotic—except 4i(p,q). (a) First and (b) second orders The circle and triangle symbols denote the system trajectories having negative and positive incident momenta p (t = 0) on the S(qi = 0), and the open and filled symbols denote those whose final states were predicted correctly and falsely by Eq. (11), respectively [45].

Peculiar particle velocity, 19 Pendulum problem, 382 Periodicity conditions, 377 Perturbed solution, 344 Pessimism-optimism rule, 316 Petermann, A., 723 Peterson, W., 212 Phase plane, 323 "Phase portrait, 336 Phase space, 13 Photons, 547... 

This function is normahzed to take the unit value for 0 = 2n. For vanishing wavenumber, the cumulative function is equal to Fk Q) = 0/(2ti), which is the cumulative function of the microcanonical uniform distribution in phase space. For nonvanishing wavenumbers, the cumulative function becomes complex. These cumulative functions typically form fractal curves in the complex plane (ReF, ImF ). Their Hausdorff dimension Du can be calculated as follows. We can decompose the phase space into cells labeled by co and represent the trajectories by the sequence m = ( o i 2 n-i of cells visited at regular time interval 0, x, 2x,..., (n — l)x. The integral over the phase-space curve in Eq. (60) can be discretized into a sum over the paths a>. The weight of each path to is... 

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

The Transition State Hypothesis. The general idea that a transition state is located at a saddle point on the PES, as detailed in Section 1.3, is familiar to most organic chemists. However, the original concept of a transition state started out as something rather different. In the development of both transition state and RRKM theory, the transition state was defined as the location of a plane (actually a hyperplane) in phase space, perpendicular to the reaction coordinate. ... 

Figure 21.6. Schematic representation of the relative phase-space volumes available to reactant, transition state, and product. A plane located at the most constricted place has the highest prohahility of being crossed only once by a molecular trajectory, which is the location of the transition state.

This process is easily generalized to systems with more variables and hence higher-dimension phase spaces. In general we specify some (N — l)-dimen-sional plane in the N-dimensional phase space and perform a similar single-cycle integration. We then have N — 1 differences, which form a vector Ax which is a function of the N — 1 initial concentration xq. We wish to make all the components of Ax zero, and the appropriate Newton Raphson form is then... 

For such phase space representations with one variable, or indeed with any number of independent concentrations, there are a number of rules which the trajectories must obey. In particular, trajectories cannot cross themselves, except at singular points (the stationary states) or if they form closed orbits (such as limit cycles or some other forms we will introduce later). Also, the trajectories cannot pass over singular points. The first of these rules is perhaps most easily shown for two-dimensional systems, where we have a two-dimensional phase plane. Let us assume that the rate equations for the two independent concentrations (or concentration and temperature), x and y, can be written in the form... 

Thus the mechanism formed by steps (l)-(4) can be called the simplest catalytic oscillator. [Detailed parametric analysis of model (35) was recently provided by Khibnik et al. [234]. The two-parametric plane (k2, k 4/k4) was divided into 23 regions which correspond to various types of phase portraits.] Its structure consists of the simplest catalytic trigger (8) and linear "buffer , step (4). The latter permits us to obtain in the three-dimensional phase space oscillations between two stable branches of the S-shaped kinetic characteristics z(q) for the adsorption mechanism (l)-(3). The reversible reaction (4) can be interpreted as a slow reversible poisoning (blocking) of... 

The asymptotics discussed above force both the concentrations of the substrate a = c- r - s - 2[RS] and the product rs to vanish ultimately. These two conditions define a line of fixed points in the three-dimensional r - s - [RS] phase space. If the initial state has a prejudice to R enantiomer such as ro > s(l, then the system ends up on a fixed line r + 2[RS] = c on a s = 0 plane, as shown in Fig. 5a. Otherwise with ro < so> the system flows to another fixed... 

To compare reactions with different time constants it is useful to plot them as trajectories in a multi-dimensional phase space whose coordinates are the species concentrations and the temperature. Fig. 2 shows trajectories projected onto the temperature vs. [r] plane for reactions with identical initial fuel and air concentrations but different initial radical concentrations and temperature. Trajectories beginning at the left had no initial radicals, and the trajectory starting at 1200 K is represented in Fig. 1. The exponential increase of [r] to [r]e is isothermal so it appears horizontal in Fig. 2. The knee of the curve represents the relatively flat portion of Fig. 1 where [r] is approximately [R]e. As the temperature increases [r] remains approximately equal to [R]e, which lies to the left of the dashed line due to consumption of fuel and oxygen. 

In analogy with the one-dimensional analysis, the Jj are defined over complete periods of the orbit in the (qj,Pj) plane, Jj = ptdq j. If one of the separation coordinates is cyclic, its conjugate momentum is constant. The corresponding orbit in the (qj,Pj) plane of phase space is a horizontal straight line, which may be considered as the limiting case of rotational periodicity, for which the cyclic qj always has a natural period of 2-k, and Jj = 2irpj for all cyclic variables. 

Pharmacokinetic studies are in general less variable than pharmacodynamic studies. This is so since simpler dynamics are associated with pharmacokinetic processes. According to van Rossum and de Bie [234], the phase space of a pharmacokinetic system is dominated by a point attractor since the drug leaves the body, i.e., the plasma drug concentration tends to zero. Even when the system is as simple as that, tools from dynamic systems theory are still useful. When a system has only one variable a plot referred to as a phase plane can be used to study its behavior. The phase plane is constructed by plotting the variable against its derivative. The most classical, quoted even in textbooks, phase plane is the c (f) vs. c (t) plot of the ubiquitous Michaelis-Menten kinetics. In the pharmaceutical literature the phase plane plot has been used by Dokoumetzidis and Macheras [235] for the discernment of absorption kinetics, Figure 6.21. The same type of plot has been used for the estimation of the elimination rate constant [236]. 

Figure 4 shows a schematic portrait of the phase space flows (denoted by arrows) in the ( i(p, q),Pi(p, q)) plane and the caricature of the corresponding... 

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