Observables time evolution

There is also another way the mesoscopic time evolution Equation (55) can be introduced. We collect a list of well-established (i.e., well tested with experimental observations) time evolution equations on many different levels of description and try to identify their common features. This is indeed the way the time evolution Equation (55) has been first introduced. The Hamiltonian structure of the nondissipative part has been discovered first in the context of hydrodynamics by Clebsch (1895). Equations of the type (55) have started to appear in Dzyaloshinskii and Volovick (1980) and later in... 

Experimental observations of the time evolution of externally unforced macroscopic systems on the level meSo l show that the level eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if Cmeso is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here Ath) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for Ath (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description (i) Mslow/ (ii) Aneso 2 -> Ath, (ui) Aneso l -> Aneso 2, and (iv) Aneso i —> Aneso 2 —> Ath- The first two are the same as (111). We now turn our attention to the third one, that is,... 

Apart from nitrophenols, 4-nitrocatechol and nitrobenzoquinone have also been detected as nitro derivatives [54,79,100]. They are secondary photoproducts and are thought to originate from the nitration of catechol and hydroquinone, in the latter case followed by the oxidation of nitrohy-droquinone [54,100]. 4-Nitrocatechol might in principle derive from catechol nitration or from 4-nitrophenol hydroxylation. However, the conversion of 4-nitrophenol into 4-nitrocatechol upon nitrate photolysis is rather limited [109] and cannot account for the observed time evolution starting from phenol [54]. 

Dielectric friction is the measure of the dynamic interaction of a charged or dipolar solute molecule with the surrounding polar solvent molecules. This concept has been applied, by Hynes et al. [339] and others [486], to solvent- and time-dependent fluorescence shifts resulting from the electronic absorption by a solute in polar solvents. If the solvent molecules are strongly coupled to the charge distribution in ground- and excited-state molecules, the relatively slow solvent reorientation can lead to an observable time evolution of the fluorescence spectrum in the nano- to picosecond range. This time-dependent fluorescence (TDF) has been theoretically analysed in terms of dynamic... 

The observed time evolution and the decay of the photoproduct absorptions by photochemical and/or thermal reactions in the TS diacetylene crystals are described quantitatively by Eqs. (15) to (18). This has been demonstrated by Hersel Neumann , and Niederwald Some examples of calculated curves are shown in Figs. 5, 10, and 12. 

In ZF-/tSR the longitudinal muon spin relaxation function G t) is directly deduced from the time-differential measurement of the forward/backward muon decay asymmetry, without any disturbance of the spin-glass system by an external field. (No depolarization of the muon spin means G = l, complete depolarization G =0.) The observed time evolution G (t) of muon-spin polarization reflects amplitudes, randomness, and fluctuations of local magnetic fields at muon sites in the specimen. There appear two essential problems in analyzing pSR experiments on spin glasses (i) One has to make model assumptions about the shape of G (t) (ii) Any relaxation slower than 10 s appears as a static component in pSR (lifetime of the muon is = 2.2 x 10 s). 

The later time evolution is shown in Figrne A3.13.7 between 90 and 100 fs, and m Figrne A3.13.8, between 390 and 400 fs, after the beginning of the excitation (time step t j)- Tln-ee observations are readily made first,... 

Quantum Cellular Automata (QCA) in order to address the possibly very fundamental role CA-like dynamics may play in the microphysical domain, some form of quantum dynamical generalization to the basic rule structure must be considered. One way to do this is to replace the usual time evolution of what may now be called classical site values ct, by unitary transitions between fe-component complex probability- amplitude states, ct > - defined in sncli a way as to permit superposition of states. As is standard in quantum mechanics, the absolute square of these amplitudes is then interpreted to give the probability of observing the corresponding classical value. Two indepcuidently defined models - both of which exhibit much of the typically quantum behavior observed in real systems are discussed in chapter 8.2,... 

Previous reports on FMSZ catalysts have indicated that, in the absence of added H2, the isomerization activity exhibited a typical pattern when measured as a function of time on stream [8, 9], In all cases, the initial activity was very low, but as the reaction proceeded, the conversion slowly increased, reached a maximum, and then started to decrease. In a recent paper [7], we described the time evolution in terms of a simple mathematical model that includes induction and deactivation periods This model predicts the existence of two types of sites with different reactivity and stability. One type of site was responsible for most of the activity observed during the first few minutes on stream, but it rapidly deactivated. For the second type of site, both, the induction and deactivation processes, were significantly slower We proposed that the observed induction periods were due to the formation and accumulation of reaction intermediates that participate in the inter-molecular step described above. Here, we present new evidence to support this hypothesis for the particular case of Ni-promoted catalysts. 

Subsequently, after about 400 s the nitrogen concentration starts to decrease and, at the same time, evolution of hydrogen in the gas phase and formation of ammonia were observed. Ammonia is by far the most important by-product of the reduction of NO adsorbed species over LNT systems, as also reported by several authors [11,27,50,51],... 

In the case of PE without NA, only the scattering intensity of stacked lamellae can be observed, so the time evolution of the LI and L2 reflections can be observed easily. In the case of PE with NA, it will be shown that qn is... 

After obtaining the time evolution of the LI reflection of PE with NA, we can separate out the scattering intensity of the isolated nuclei at from the observed scattering intensity. Hereafter the range [Pg.147]

Time evolutions of Ix 0bs(< >t) of the mixture of PE with NA and that of PE without NA are shown in Figs. 7 and 8, respectively. They were observed at the same degree of supercooling AT as the optical observation shown in Fig. 6. The induction time observed by optical microscopy (ti(OM)) was... 

The time evolution of the integrated scattering intensity of Ix(q yt) and Ix(ql> t) = Jx(L2) of the mixture of PE with NA is shown in Fig. 9. Ix(qn, t) starts increasing soon after quenching and increased quickly up to ri(OM) = 2.5 min with the increase of time. The rate of increase slowed after 6 min. This suggests that the formation of isolated nuclei during the induction time has been observed for the first time. 

Fig. 12 Comparison of time evolution of 7x(Nucleus) for PE with NA and that of 7X(L1) for PE without NA. Nucleation during induction period was clearly confirmed. Nucleation was observed only when NA was mixed. Without mixing NA, only formation of stacked lamellae was observed... 

This problem is very important, but it is extremely difficult to make SAXS measurements at higher temperatures because the induction period becomes too short to observe the time evolution of SAXS intensities. For example, as was seen in Sect. 2.2, the induction period was only 100 s when the PET glass was crystallized even at 115 °C, 40 K higher than Tg, where a detailed analysis of the SAXS data was impossible. Of course, as the crystallization temperature approaches the melting temperature, the induction period is expected to become longer. However, as will be shown below, no characteristic peaks of SD could be detected in SAXS curves either. This is probably because the crystallization temperature was not in the unstable state, or the characteristic wavelength was much larger compared with the lower resolution limit of... 

Figure 25 shows the time evolution of the difference SAXS intensity which was observed in situ when a PET sample was crystallized by cooling down from... 

The probability to observe a particular sequence of states depends on the distribution of the initial conditions and the dynamical rule describing the time evolution... 

Fig. 4. A summary of the time evolution of primordial 4He abundance determinations (mass fraction Yp) from observations of metal-poor, extragalactic Hu regions (see the text for references). The solid horizontal line is the SBBN-predicted 4He abundance expected for the WMAP (and/or D) inferred baryon density. The two dashed lines show the la uncertainty in this prediction.

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