Scaling functions momentum

To summarize, strict e-expansion a priori seems to yield unambiguous results. Closer inspection, however, reveals that in low order calculations considerable ambiguity is hidden in the definition of the physical observables used as variables or chosen to calculate. What is worse, the e-expansion does not incorporate relevant physical ideas predicting the behavior outside the small momentum range or beyond the dilute limit. In particular, it does not give a reasonable form for crossover scaling functions. On the other hand, it can be used to calculate well-defined critical ratios, which are a function of dimensionality only, Even then, however, the precise definition of the ratio matters,... 

We must however add a word of caution. The method as sketched here has been established to deal with the small momentum behavior of the scaling functions. For large momenta we may run into problems in that the subleading momentum dependence is not treated correctly. On the quantitative level however this does not seriously invalidate our results. We will discuss this aspect further in Sect. 15.2. See also Sect. 13.2.3,... 

The calculation of the resistance coefficients can be accomplished in the frame of the Monin-Obukhov similarity theory (Monin and Yaglom, 1971). The genuine flux quantities are the friction velocity u and the scale functions 0 and referring to temperature and humidity. The turbulent momentum flux f, the sensible heat flux the mass flux from evaporation and condensation and the corresponding latent heat flux are... 

When using LES, the time-dependent three-dimensional momentum and continuity are solved for. A subgrid turbulence model is used to mode the turbulent scales that are smaller than the cells. Instead of the traditional time averaging, the equations for using LES are filtered in space, and is a function of space and time. 

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

For different momentum transfers the dynamic structure factors are predicted to collapse to one master curve, if they are represented as a function of the Rouse variable. This property is a consequence of the fact that the Rouse model does not contain any particular length scale. In addition, it should be mentioned that Z2/ or the equivalent quantity W/4 is the only adjustable parameter when Rouse dynamics are studied by NSE. 

Fig. 7. Characteristic relaxation rate for the Rouse relaxation in polyisoprene as a function of momentum transfer. The insert shows the scaling behavior of the dynamic structure factor as a function of the Rouse variable. The different symbols correspond to different Q-values. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time is defined as the time it takes for the incoherent (circles) or coherent (squares) intermediate scattering function at a momentum transfer given by the position of the amorphous halo (q — 1.4A-1) to decay to a value of 0.3. The full line is a fit using a VF law with the Vogel-Fulcher temperature T0 fixed to a value obtained from the temperature dependence of the dielectric a relaxation in PB. The dashed line is a superposition of two Arrhenius laws (see text).

Fig. 4.14 Results on fully protonated PIB by means of NSE [147]. a Time evolution of the self-correlation function at the Q-values indicated and 390 K. Lines are the resulting KWW fit curves (Eq. 4.9). b Momentum transfer dependence of the characteristic time of the KWW functions describing Sseif(Q,t) at 335 K (circles), 365 K (squares) and 390 K (triangles). In the scaling representation (lower part) the 335 K and 390 K data have been shifted to the reference temperature 365 K applying a shift factor corresponding to an activation energy of 0.43 eV. Solid (dotted) lines through the points represent (q-2 power laws. Full...

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

Figure 4. The average change in the squared value of the classical orbital angular momentum ((A/. )) and the standard deviation of the A/. distribution

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