Time-evolution plots

Assuming that an equilibrium is now well established, the simulation may be restarted (not newly started) to begin with the sampling of structural and thermodynamic data. In our model case, data acquisition was performed for 3 ns (trajectory data plot not shown). For the production phase, also, the time evolution of the variables mentioned above should be monitored to detect stability problems or con-... 

Consider the evolution of initial states in which only a single site has nonzero amplitude. Figure 8.11 shows grcy-scalc plots (or probability maps) for a system with 125 sites evolving according to equation 8.54 with 6 = 6 = dc = 0.05, 2 and 20 in eac.h case the number of time steps plotted is 600. We make a few general observations. 

Figure 2.11 (a) Waterfall plot of the time evolution of Raman spectra ofadenine polycrystal, (b) Five spectra taken from (a). The exposure time for each spectrum is 10s. 

The time evolution of the absorption intensity of each component is plotted in Fig. 19. Since the conformation in the iPS crystal is 3/1 helix (TG)3, two GTGT bands at 567 and 586 cm-1 may be assigned to a 3/1 helical conformation, which corresponds to the stiff segments. Here, it should be noted... 

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

The first term on the right-hand side is the expected value of the scalar Laplacian conditioned on the scalars having values r//, 33 An example of the time evolution of the conditional scalar Laplacian, corresponding to the scalar PDF in Fig. 1.11, is plotted in Fig. 1.12 for an initially non-premixed inert-scalar field. The closure of the conditional scalar Laplacian is discussed in Chapter 6. For the time being, it suffices to note the similarity between (1.36) and the IEM model, (1.16). Indeed, the IEM model is a closure for the conditional scalar Laplacian, i.e.,... 

The short program Lorenz.m calculates the concentrations for A, B and C for the initial conditions. c0=[l l 20]. Figure 3-37 displays the trajectories in a fashion that is not common in chemical kinetics. It is a plot of the time evolution of the values of A vs. B vs. C (see also Figure 3-35). Most readers will recognise the characteristic butterfly shape of the trajectory. The important aspect is that, in contrast to Figure 3-35, the trajectory is different each time. This time, it is not the effect of numerical errors but an essential aspect of the outcome. Even if the starting values for A, B and C are away from the butterfly , the trajectory moves quickly into it it is attracted by it and thus the name, Lorenz attractor. 

The integrated form of the foreing funetion/(t) deseribes the time evolution of the system s transition between these two states (shown in the following plots as the dashed lines the adjustment of the chemical system in response to the altered rate constants is shown as the solid lines). 

The immediate question is then how is this compatible with the arguments concerning sensitivity of the system to the value of concentration at the minimum and the expected related positive rectification To answer, we have to examine the detailed time evolution of the minimal electrolyte concentration Cm n(t) (the interface concentration in the electro-neutral picture) during one period. Bear in mind that, since at the plateau of the VC curve practically all of the system s resistance is concentrated at the location where the concentration is at its minimum, the electric current in the system is proportional to Cmjn(f) V(t). In Fig. 5.4.4 we present the calculated time plots of Cm[n = C(t, 1), V(t) during one period for / = 1, A = 10, Vcr= 15. 

The time evolution of the cathodic limiting current (Eq. 2.147) has been plotted in Fig. 2.15 together with that obtained for a planar electrode (Eq. 2.28) and the constant steady-state limiting current for a spherical electrode given by... 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

Figure 28 Two-dimensional contour plots of the calculated PES for die CT2O3 (0 00 l)-CO system. 4>—X dependence at ground state (bottom) and at die excited a3 n state (top) with a snapshot of die wave packet after a time evolution of 50 fs. Inset cluster model, only two Cr ions are shown [79].

Figure 7. Time evolution of the standard deviation of the length (

Chronodeflectogram — A deflection signal where the deviation angle (6) of a probe laser beam is plotted as a function of time. It is also called chronodeflectomet-ric profile or PBD transient and is usually characterized by the presence of the - PBD maximum (or minimum) during the time evolution of the deflection signal [i]. See also - chronodeflectometry. 

Chronodeflectometry (CD) — A - probe beam deflection method in which a - potential pulse is applied, usually from a potential where there is no reaction, to one where the reaction evolves completely to the formation of products. The time evolution of the deflection signal is sampled and plotted as a function of time, which is named chronodeflectogram [i]. 

Figure 25 (upper plot) A schematic plot of the enantiodiscriminator. The three levels of each enantiomer are resonantly coupled by three fields. The dipole moments of the two enantiomers have opposite signs, (middle plot) The time evolution of the population of the three levels. The D and L enantiomers start in the 1) state. At the end of the process one enantiomer is found in the 3) state and the other in the 1) state, (lower plot) The time-dependence of the eigenvalues of the Hamiltonian of Eq. (73). The population initially follows the E0) dark state. At t rthe population crosses over diabatically to ) for one enantiomer and to E+) for the other. 

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