Phase 9 point

Fig. XVII-22. Isosteric heats of adsorption for Kr on graphitized carbon black. Solid line calculated from isotherms at 110.14, 114.14, and 117.14 K dashed line calculated from isotherms at 122.02, 125.05, and 129.00 K. Point A reflects the transition from a fluid to an in-registry solid phase points B and C relate to the transition from the in-registry to and out-of-registry solid phase. The normal monolayer point is about 124 mol/g. [Reprinted with permission from T. P. Vo and T. Fort, Jr., J. Phys. Chem., 91, 6638 (1987) (Ref. 131). Copyright 1987, American Chemical Society.]...

The statement of the mixing condition is equivalent to the followhig if Q and R are arbitrary regions in. S, and an ensemble is initially distributed imifomily over Q, then the fraction of members of the ensemble with phase points in R at time t will approach a limit as t —> co, and this limit equals the fraction of area of. S occupied by... 

Figure C3.6.7(a) shows tire u= 0 and i )= 0 nullclines of tliis system along witli trajectories corresponding to sub-and super-tlireshold excitations. The trajectory arising from tire sub-tlireshold perturbation quickly relaxes back to tire stable fixed point. Three stages can be identified in tire trajectory resulting from tire super-tlireshold perturbation an excited stage where tire phase point quickly evolves far from tire fixed point, a refractory stage where tire system relaxes back to tire stable state and is not susceptible to additional perturbation and tire resting state where tire system again resides at tire stable fixed point.

Of all the characteristic points in the phase diagram, the composition of the middle phase is most sensitive to temperature. Point M moves in an arc between the composition of the bottom phase (point B) at and the composition of the top phase (point T) at reaching its maximum surfactant concentration near T = - -T )/2. (Points B and Tmove by much smaller amounts, also.) The complete nonionic-amphiphile—oh—water—temperature... 

Fig. 16. Gibbs energy-temperature diagram if FCC and ECC are present in the system. Ai-isotropic (undeformed) melt, A2-deformed melt (nematic phase) points 1 and 4 - melting temperatures of FCC and ECC under unconstrained conditions (transition into isotropic melt) points V and 2 -melting temperatures of FCC and ECC under isometric conditions (transition into nematic phase), point 3 - melting temperature of nematic phase (transition into isotropic melt but not completely randomized)...

Figure 15. Left-. Geometry of the surface 6 = 0 in Eq. (46) with fixed total angular momenta S and N. Properties of the special points A, B, C, and D are listed in Table I. All other permissible classical phase points lie on or inside the surface of the rounded tetrahedron. Right. Critical section at J. Continuous lines are energy contours for y = 0.5 and N/S = 4. Dashed lines are tangents to the section at D. Axes correspond to normalized coordinates, /NS and K JiN + S). Taken from Ref. [2] with permission of Elsevier.

Consider a system of N particles with masses m in a volume V = L3. Particle i has position r, and velocity v, and the phase point describing the microscopic state of the system is /e (r, v ) = (ri, r2,..., rN, vi, V2,..., v v). We assume that the particles comprising the system undergo collisions that occur at discrete-time intervals x and free stream between such collisions. If the position of particle i at time t is r, its position at time t + x is... 

In multiparticle collisions the same rotation operator is applied to each particle in the cell c but every cell in the system is assigned a different rotation operator so collisions in different cells are independent of each other. As a result of free streaming and collision, if the system phase point was (r, N) at time t, it is (r, v ) at time t + x. 

In order to cast the expression for K(k) into a form that is convenient for its evaluation, it is useful to establish several relations first. To this end we let S(xN, t) denote the value of the phase point at time x whose value at time zero was x V. We then have... 

Phase, Phase Point 0 (degree) [Pg.41]

The "fluid" formed from the phase points is thus incompressible. Furthermore,... 

For an isolated system based on the microcanonical ensemble with phase points equally distributed in a narrow energy range, all systems exhibit the same long-term behaviour and spend the same fraction wt of time in different elementary regions. For a total number N of systems at time t,... 

Over a long time r, the number of phase points traversed is... 

The simulation result (Figure 4) shows that when two initial conditions are very close, after a dimensionless time of 40 units the concentration of reactant A and the reactor temperature are completely different. This means that the system has a chaotic behavior and their d3mamical states diverge from each other very quickly, i.e. the system has high sensitivity to initial conditions. This separation increases with time and the exponential divergence of adjacent phase points has a very important consequence for the chaotic attractor, i.e. 

In the new series, each diagram contains points, and each point has associated with it a phase-point variable and a time. (Adapted from Andersen, 2003)... 

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

Fig. 16a symmetric limit cycles for the second-harmonic mode (GCL) and in Fig. 16b, an nonsymmetric phase portrait example for 7) = 0.5 for BCL. In both cases the phase point settles down into a closed-loop trajectory, although not earlier than about x > 200. An intricate limit cycle is usually related to multiperiod oscillations. For example, the cycle in Fig. 16a corresponds to five-period oscillations of the fundamental and SHG modes intensity, and the phase portrait in Fig. 16b resembles the four-period oscillations (see Fig. 17). Generally, for 7) > 0.5, we observe many different multiperiod (even 12-period) oscillations in intensity and a rich variety of phase portraits.

The only difference is that for the harmonic oscillator the phase point draws the ellipse with the frequency too, whereas for the Kerr oscillator with the frequency, = o>o[l + e(pq + 0) 2)]. The frequency depends on the initial conditions, which is a feature typical of nonlinear conservative systems [143]. 

The single Kerr anharmonic oscillator has one more interesting feature. It is obvious that for Cj = 0 and y- = 0, the Kerr oscillator becomes a simple linear oscillator that in the case of a resonance 00, = (Do manifests a primitive instability in the phase space the phase point draws an expanding spiral. On adding the Kerr nonlinearity, the linear unstable system becomes highly chaotic. For example, putting A t = 200, (D (Dq 1, i = 0.1 and yj = 0, the spectrum of Lyapunov exponents for the first oscillator is 0.20,0, —0.20 1. However, the system does not remain chaotic if we add a small damping. For example, if yj = 0.05, then the spectrum of Lyapunov exponents has the form 0.00, 0.03, 0.12 1, which indicates a limit cycle. 

Over time, a dynamical system maps out a trajectory in phase space. The trajectory is the curve formed by the phase points the system passes through. We will return to consider this dynamic behavior in Section 3.2.2. 

If we start a system at some reasonable (i.e., low-energy) phase point, its energy-conserving e volution over time (i.e., its trajectory) seems likely to sample relevant regions of phase space. 

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