Correlation length time-dependent

The case m = 6 corresponds to the most long-range dipole-dipole interaction. Define how the time-dependent correlation length = o (subscript O indicates that the linear approximation is employed) ... 

It has been postulated that after some initial transient time following a quench, H x,t) should change its shape self-similarly with time. In this section, we examine what comes out of this postulate. The self-similarity of H (x, t) is mathematically equivalent to the condition that there exists a single time-dependent correlation length (t) which satisfies the relation... 

We now address the dynamic structure factor which incorporates all time-dependent correlations of segments along the chain. While the early MD-sim-ulations of Kremer and Grest did very well with the msd, the dynamic structure factor was only poorly described. Figure 3.25b displays a comparison with NSE results on PEP and PE, where the simulation results were mapped to the experiment in terms of time units measured by (F= f/r ) and length scales measured by the tube diameter d(Q = Qld) [50]. 

When a binary mixture with the critical composition (f>c is suddenly brought to a temperature T inside the spinodal but sufficiently close to the critical temperature Tc, we call the operation a very shallow critical quench (VSCQ). During the spinodal decomposition after a VSCQ the mixture stays in the vicinity of the critical state so that we may expect its dynamic properties at different T to be scaled by a T-dependent correlation length c(T) and a T-dependent time constant tc T) which characterize near-critical mixtures. The expected scaling laws for k t, T) and 1 (1, T) (- I t, k) at its maximum) are... 

Fig. 4 illustrates the time-dependence of the length of top s water column in conical capillary of the dimensions R = 15 pm and lo =310 pm at temperature T = 22°C. Experimental data for the top s column are approximated by the formula (11). The value of A is selected under the requirement to ensure optimum correlation between experimental and theoretical data. It gives Ae =3,810 J. One can see that there is satisfactory correlation between experimental and theoretical dependencies. Moreover, the value Ae has the same order of magnitude as Hamaker constant Ah. But just Ah describes one of the main components of disjoining pressure IT [13]. It confirms the rightness of our physical arguments, described above, to explain the mechanism of two-side liquid penetration into dead-end capillaries. 

The relaxation rates calculated from Eq. (15) are smaller than the measured ones at low field, while they are larger at high field. OST is thus obviously unable to match the experimental results. However, water protons actually diffuse around ferrihydrite and akaganeite particles and there is no reason to believe that the contribution to the rate from this diffusion would not be quadratic with the external field. This contribution is not observed, probably because the coefficient of the quadratic dependence with the field is smaller than predicted. This could be explained by an erroneous definition of the correlation length in OST, this length is the particle radius, whilst the right definition should be the mean distance between random defects of the crystal. This correlation time would then be significantly reduced, hence the contribution to the relaxation rate. 

The actual time behaviour of F(t), shown in equation (4.1.41), depends on the choice of cr(r). This quantity F(t) could be presented in a more universal and transparent form in terms of the correlation length . 

To simplify mathematical manipulations, let us consider now the case of equal diffusion coefficients, Da = D, in which case the similar correlation functions just coincide, Xv r),T) = X(t),t). Taking into account the definition of correlation length Id = VDt, where D = Da + D = 2D a, as well as time-dependence of new variables r) and r, one gets from (5.1.2) to (5.1.4) a set of equations... 

The optimal values of the parameters a/reg and a time depend on the sampling of the time-frequency domain. For the values used in our implementation, A z = 0.2 Bark and A t = 20 ms (total window length is about 40 ms), the optimal values of the parameters in the model were found to be a freq = 0.8, a time = 0.6 and y = 0.04. The dependence of the correlation on the time-domain masking parameter OQj me turned out to be small. 

The bond length changes determined from pre-resonance Raman spectra, electronic spectra and time dependent theory provide a detailed picture of the results of bonding changes caused by populating excited electronic states. There is a direct but not linear correlation between bond length changes and the... 

Driven by a Boltzmann thermal-energy source kT and measured in kT units, coupled by a number of effective mobile ionic charges rs from Gibbs, the monopole-monopole correlation force is screened by Xnebye across the length 21, back and forth between point particles. At the same time its power-law dependence on length is measured in the natural thermal unit >.Bj. Boltzmann, Gibbs, Debye, Bjerrum—all at the same time. Can it get any prettier ... 

Analogously to the analysis presented in Section 10.3.1.1.2 for the gas phase, Nieber and Doltsinis [64] have attempted to establish possible correlations between the time-dependent surface hopping parameter, P10, and certain geometric variables as well as the S0 - Sr energy gap. The hopping parameter P10 appears to be modulated by variations in the C(5)C(6) and C(4)C(5) bond lengths. Moreover, the oscillations in the time derivative of the S0 - energy gap are seen to match well those of P10. Hence, there does not seem to be any qualitative difference in the mechanism of nonradiative decay in solution compared to the gas phase. 

We are in the process of examining time dependent properties such as the diffusion coefficient and the relaxation times of various correlation functions as a function of both chain length and density. Following the work of Kirkwood (24) and Rouse (24) we assume that the velocity of the polymer is proportional to the forces acting on it at any time this is the high-viscosity limit in which inertial terms are neglected. Neglecting also hydrodynamic forces, we then have for the velocity of the jth bead at time t... 

The actual time behaviour of F t), shown in equation (4.1.41), depends on the choice of cr r). This quantity F t) could be presented in a more universal and transparent form in terms of the correlation length Let us consider as an illustration the multipole interaction which is observed in many solids and liquids due to the interaction of the electronically excited donors D with acceptors A resulting in an energy transfer D + A D + A + A + hu [16, 17]. Its probability (per unit time) is... 

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