Phase space circle

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

Fig. 18 Phase space of PI-fc-PS-fc-PEO in vicinity of ODT. Filled and open circles-. ordered and disordered states, respectively, within experimental temperature range 100 < T/° C< 225. Outlined areas compositions with two- and three-domain lamellae (identified by sketches) shaded regions three network phases, core-shell double gyroid (Q230), orthorhombic (O70), and alternating gyroid (Q214). Overlap of latter two phase boundaries indicates high- and low-temperature occurrence, respectively, of each phase. Dashed line condition tfin = 0peo associated with symmetric PI-fc-PS-fc-PEO molecules. From [75]. Copyright 2004 American Chemical Society...

Fig. 29. Phase diagram of the model Eq. (22) for coadsorption of two kinds of atoms in the temperature-coverage space. Circles indicate a second-order phase transition, while crosses indicate first-order transitions. Point A is believed to be a tricritical point and point B a bicritical point. The dashed curve shows the boundary from the Blume-Capel model on a square lattice with a nearest-neighbor coupling equal to 7 in the present model (for - 0 Eq. (22) reduces to this model), only the ordered phase I then occurs. From Lee and Landau. )...

Certain aspects of this phase space trajectory merit attention. We noted above that a phase space trajectory cannot cross itself. However, it can be periodic, which is to say it can trace out the same path again and again the harmonic oscillator example is periodic. Note that the complete set of all harmonic oscillator trajectories, which would completely fill the corresponding two-dimensional phase space, is composed of concentric ovals (concentric circles if we were to choose the momentum metric to be (mk) 1/2 times the position metric). Thus, as required, these (periodic) trajectories do not cross one another. 

Figure 21. Structure of the phase space of the Lorenz system. An escape trajectory measured by numerical simulation is indicated by the filled circles. The trajectory of the Lorenz attractor is shown by a thin line the separatrixes T and T2 by dashed lines [182].

The trajectories of dissipative dynamic systems, in the long run, are confined in a subset of the phase space, which is called an attractor [32], i.e., the set of points in phase space where the trajectories converge. An attractor is usually an object of lower dimension than the entire phase space (a point, a circle, a torus, etc.). For example, a multidimensional phase space may have a point attractor (dimension 0), which means that the asymptotic behavior of the system is an equilibrium point, or a limit cycle (dimension 1), which corresponds to periodic behavior, i.e., an oscillation. Schematic representations for the point, the limit cycle, and the torus attractors, are depicted in Figure 3.2. The point attractor is pictured on the left regardless of the initial conditions, the system ends up in the same equilibrium point. In the middle, a limit cycle is shown the system always ends up doing a specific oscillation. The torus attractor on the right is the 2-dimensional equivalent of a circle. In fact, a circle can be called a 1-torus,... 

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].

One more important ingredient in discussing the transport process in mixed phase space is so-called cantori. They are invariant sets in which the motion has an irrational frequency. They resemble invariant circles, but they have an infinite number of gaps in them. The existence was proposed by Percival [30] and Aubry [31]. They gave an explicit example, and a proof of their existence has been given afterwards [32-34]. 

Since the system is 27r-periodic in 0, it may be considered as a vector field on a cylinder. (See Section 6,1 for another vector field on a cylinder.) The x-axis runs along the cylinder, and the 0-axis wraps around it. Note that the cylindrical phase space is finite, with edges given by the circles x = 0 and x = 1. 

Figure 9. Snapshots of the phase space distribution (PSD) obtained from classical trajectory simulations based on the fewest-switches surface-hopping algorithm of a 50 K initial canonical ensemble [46], Na atoms are indicated by black circles, and F atoms are indicated by gray crosses. Dynamics on the hrst excited state starting at the Cj structure (t = 0 fs) over the structure with broken Na-Na bond t = 90 fs) and subsequently over broken ionic Na-F bond (t = 220 fs) toward the conical intersection region (t = 400 fs), Dynamics on the ground state after branching of the PSD from the hrst excited state leads to strong spatial delocalization (t = 600 fs). The C2v isomer can be identihed at 800 fs in the center-of-mass distribution. See color insert.

It is clear from (28) that g(a) is always positive, since p is a positive definite operator. For a coherent state ao), g(a) = (l/ji)exp(— a — ao 2) is a Gaussian in the phase space Re a, Im a which is centered at a0. The section of this function, which is a circle, represents isotropic noise in the coherent state (the same as for the vacuum). The anisotropy introduced by squeezed states means a deformation of the circle into an ellipse or another shape. 

Asymptotic stability never appears, because it is not possible for the eigenvalues A3, A4 to be both inside the unit circle. This is also a consequence of the fact that the volume in phase space is conserved. 

Although the phase space trajectories appear as simple curves on the two-dimensional Iz,ip phase space diagram (the 0 coordinate is suppressed) most trajectories are actually quasiperiodic. The actual trajectories he on the 2-dimensional surface of a 3-dimensional invariant torus in 4-dimensional phase space. Fig. 9.14 shows such a torus. Any point on the surface of the torus is specified by two angles, 0 and. The 0 and circuits about the torus are shown, respectively, as large and small diameter circles. The diameter of the 0... 

Figure 7.4 Calculated k( ) curves for the bromobenzene ion. The open circles are obtained by lowering the last five vibrational frequencies to 113 cm" in the transition state. The PST (phase space theory) rate constant had the lowest two frequencies replaced by free rotors, while in the other three lines all transition state frequencies were multiplied by the indicated factor. The E and the parent ion vibrational frequencies were the same for all calculations. Note the different slopes.

Fig. 25. The temperature dependence of the layer spacing for the dodecyl (C12) (filled circles), tetradecyl (C14) (open circles), and the hexadecyl (Cie) (triangles) homologues of the (R)-l-methylheptyl 4 -(4-n-alkoxyphenylpropioloyloxy)biphenyl-4-carboxylates (nPlM7 s). The inset shows scans through the layer peak for the tetradecyloxy homologue at temperatures within the smectic C phase (filled circles T = 82.8 °C), the TGBA phase (open circles T = 93.3 °C), and just into the isotropic liquid (triangles T = 97.4 °C)...

Fig. 10. Snapshot (t = tof spiral wave solution of Oregonator equations predicted by Eq. (23) and Fig. III.4. Compare with Fig. 8. At fixed radius (r = rj l/k), x(rj, 6,t ) = x(0+constant) that is, the dashed circle in real space maps onto the limit cycle in phase space. At fixed angle (0 = 0 ), x(r, 0, t ) = x(constant - kr) for kr l the dashed line between (rj, 0 ) and (r + 277/k,0j ) maps onto the limit cycle as well. Points 0 m in real space map onto points -H) in the phase plane.

FIGURE 8.3 The possible movements of the bistable reaction system through the phase space ([Y]ss, A) in the vicinity of the bifurcation points and the corresponding time evolution [Y]ss- The empty circle denotes the starting state of the system [32]. 

Figure 5.9 Schematic diagram of a quenching experiment in phase space. F is the steady state the circle is the limit cycle, with / the point at which the system is perturbed. The quenching vector q shifts the state from / to P, which lies on the stable manifold along t. The system then moves to F. The vector d shows the result of a dilution. The radius r varies with the phase of oscillation. (Adapted from Hynne and Sorensen, 1987.)...

It is very convenient to represent the eigen solutions of (23) as trajectories in the phase space. Then solitary and level jump waves are represented by homoclinic and heteroclinic trajectories accordingly. Periodic and biperiodic solutions in phase space by circles and invariant torii are represented. 

Figure 8. Diagram showing crystallographic "d" spacings versus pressure for the different RDX solid phases. For a RDX, the normal orthorhombic pattern is shown at 2.6 GPa (squares). At 4.5 and 5.1 GPa, another orthorhombic pattern is shown (diamonds) for the 7 phase. The circles indicate the "d spacings found for the p phase at 5.6 and 6.7 GPa. All the data were measured at room temperature.

Fig. 3.23. Temperature dependence of the rate coefficient for the reaction C2HJ+H2 —> C2H + H. As discussed by Gerlich, there is some disagreement between the free jet and the ion trap experiments. While, at low temperatures, the high pressure experiments seems to indicate an increasing rate coefficient, the 10 K ion trap results proof that the rate coefficient for the abstraction reaction is much smaller than 10 cm s . A possible explanation is the fast radiative association process (fcr(p-H2) = 5 X 10 cm s ), which has been observed in the ion trap experiment (open circle and triangle). The phase space calculations (solid line) have used adjusted parameters for getting the low temperature behavior.

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