Local displacement, time

The kinetics of the nonconserved order parameter is determined by local curvature of the phase interface. Lifshitz [137] and Allen and Cahn [138] showed that in the late kinetics, when the order parameter saturates inside the domains, the coarsening is driven by local displacements of the domain walls, which move with the velocity v proportional to the local mean curvature H of the interface. According to the Lifshitz-Cahn-Allen (LCA) theory, typical time t needed to close the domain of size L(t) is t L(t)/v = L(t)/H(t), where H(t) is the characteristic curvature of the system. Thus, under the assumption that H(t) 1 /L(t), the LCA theory predicts the growth law L(t) r1 /2. The late scaling with the growth exponent n = 0.5 has been confirmed for the nonconserved systems in many 2D simulations [139-141]. 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

The abscissa is actually the Reynolds number based on local displacement thickness. In this case, only the local solution is predominant at early times. This component also disperses and decays, as can be seen from the solution at t = 100. The observed single peak at t = 0 that is due to the local solution disperses into multiple peaks- as can be noted for all subsequent times. This dispersion of solution is due to the presence of upstream prop>-agating modes and the presence of multiple harmonics for the downstream propagating modes. The adjective upstream here is to be understood with respect to the local condition of the disturbance field. More details about this dispersion mechanism and tracking of the upstream propagating modes were first discussed in Sengupta et al. (1999) and will be discussed again in the next subsection. 

The time dependence of local displacement c and displacement velocity dc/dt at the craze or crack initiation is shown schematically in Figure 1. The velocity drops to one half of its maximum value at ti/2 If one puts t = 0 at the maximum and assumes a symmetric time dependence of displacement velocity, the half width of velocity distribution at such an elementary act is 2ti/2. The Fourier transform (FT) of velocity yields the spectral distribution of the emitted acoustic burst. The intensity is a maximum at zero frequency and drops to half this value at (01/2 = 2mv /2> In the first approximation the product of the half width of velocity and frequency distribution of acoustic emission, 2ti/2 X 2vi/2, equals 0.8825. To have a substantial amount of energy available in the frequency range of 1 MHz ( —1 1/2) the displacement velocity curve vs. time must have a half width of 0.5 /xsec, i.e., the major part of the local displacement must occur within 0.5 /i.sec. The square of the maximum value of velocity times... 

It is possible to sample stochastic transition pathways by making only local displacements of trajectories. For example, a randomly chosen time shce of an existing pathway may be modified by adding a small displacement 5x to positions and momenta, xj" = xj" + x. All other time slices remain unchanged. This modification, which is local in time, gives a different but finite path probability If the displacement x is chosen from a... 

The simulated dilatations involved increasing steps of imposed dilatation on the simulation cell. To permit a detailed understanding of the dilatational response of the polymer at the atomic level the entire volume of the simulation cell was tessellated into Voronoi polyhedra at each atomic site, permitting determination of strain-increment tensor elements dcy for each site from local displacement gradients by a technique described by Mott et al. (1992). Such increments of imposed dilatation at a level of 3 x 10 were applied 100 times to obtain total system dilatations of 0.3 (Mott et al. 1993b). For eaeh dilatation increment the atomic site strain-tensor increments de were obtained for each site n. The two invariants, de", the atomic site dilatation increment, and the work-equivalent shear-strain increment, dy", were obtained from the individual increments as... 

Assuming that i and p are white noises and their correlations are local in time, the mean square displacement of a monomer is given by... 

Let < j(k) be the Klein-Gordon amplitude corresponding to a spin zero particle localized at the origin at time t = 0. Since in momentum space the space displacement operator is multiplication by exp (— tk a), the state localized at y at time t = 0 is given by exp (—ik-y) (k). This displaced state by condition (b) above must be orthogonal to (k), i.e. 

Summarizing, the in situ UV-Vis, XANES, and EXAFS studies of Bonino et al. [49] and of Prestipino et al. [50] on hydrated and anhydrous peroxo/hy-droperoxo complexes on crystalhne microporous and amorphous meso-porous titanosilicates have shown, for the first time, the equilibriiun between r] side-on and end-on complexes. The amount of water is the key factor in the equilibrium displacement. In this regard please note that, owing to the hydrophobic character of TS-1, substrates such as olefins are the dominant species in the channels. This fact assures a relatively local low concentration of water, which in turn guarantees a sufficient presence of the active end-on... 

Local reptation regime For times t > xe we have to consider curvilinear Rouse motion along the spatially fixed tube. The segment displacement described by Eq. (18) (n = m) must now take the curvilinear coordinates s along the tube into consideration. We have to distinguish two different time regimes. For (t < xR), the second part of Eq. (19) dominates - when the Rouse modes... 

For this algorithm, one can prove that detailed balance is guaranteed and the exact average of any configuration-dependent property over the accessible space is obtained. Two key issues determine the detailed balance. The first is the fact that the trial probability to pick the displacement vector Dfc to go from the fcth to the Zth e-sphere equals the trial probability to pick the displacement vector D fc for the reverse step. The second issue is that the trial probability for a local MC step that moves the walker from a point inside an e-sphere to a point outside that sphere is the same as for the reverse move i.e., (1 - / ) times what it would be in a walk restricted to local moves. 

Fig. 1. Evolution of 3He/H in the solar neighborhood, computed without extra-mixing (upper curve) and with extra-mixing in 90% or 100% of stars M < 2.5 M (lower curves). The two arrows indicate the present epoch (assuming a Galactic age of 13.7 Gyr) and the time of formation of the solar system 4.55 Gyr ago. Symbols and errorbars show the 3He/H value measured in meteorites (empty squares) Jupiter s atmosphere (errorbar) the local ionized ISM (filled triangle) the local neutral ISM (filled circle) the sample of simple Hll regions (empty circles). Data points have been slightly displaced for clarity. The He isotopic ratios has been converted into abundances relative to hydrogen assuming a universal ratio He/H= 0.1. See text for references.

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