Average propagator

Recognizing that the information transmission is often both nonuniform (since the left and right fronts do not always propagate outward from the perturbation with a single well-defined velocity), and dependent on choice of initial states, one can instead define maximal and minimal average propagation speeds, and A " " ... 

It is possible to define average propagation rate constants for copolymerization subject to a penultimate group effect as follows. 

Note that equation 88 is based on the pseudo-homopolyirerlzatlon jproacli It reduces to liie sinple ptroduct of moncmer concentration by a suitably conposltlon averaged propagation constant and the total nuntoer of active chains in the particles. 

The instantaneous direction of propagation is, of course, perpendicular to the plane defined by the instantaneous electromagnetic fields E and B. But this time-dependent direction need not be parallel to the z axis, physically defined as the direction for the average propagation of energy. Let us illustrate the point with variations of the same simple example of previous section, for additional details see Munera and Guzman [67]. 

On the other hand, the average propagation of energy along the v axis is quite different zero in the plane case (Example 1), and negative in the nonplanar Example 2. This means that the wave absorbs energy from the surroundings. As... 

Using the concept of the dynamic displacement, it is possible to rewrite Eq. (1) so as to define a very useful function, known as the average propagator (Karger and Heink, 1993) F(R, t). This function gives the average probability for any particle to have a dynamic displacement R over a time t and is given by... 

Note that for the example of simple unrestricted Brownian motion, all molecules experience an identical average propagator, irrespective of starting position, reflecting the Markov nature of the statistics. This is just one case that will be encountered in the study of molecule translational motion using NMR methods. This case is easily extended to include simple flow with common velocity v. The solution is... 

We will find it convenient to consider for the moment the special case of perfectly reflecting walls. In the long time-scale limit A a /D, the average propagator for fully restricted diffusion has a simple relationship to the pore geometry. This requirement on A, also known as the pore equilibration condition, implies that the time is sufficiently long that most molecules have collided with the walls. Under this condition the conditional probabilities are independent of starting position so that P(r, t I r, 0) reduces to p(r ), the pore molecular density function. 

In consequence, the averaged propagator P(R,t) becomes an autocorrelation function of p(r ). 

The quantity f q) is the diffraction pattern, and a R) is the Patterson function (cf. Fig. 5.4.3). This interpretation relates to the description of PFG NMR in terms of probability densities or average propagators. In fact, the condition (5.4.20) defines a cross-section in the 2D k space spanned by fei and k2, so that by the projection-crosssection theorem (5.4.12) the Patterson function can be interpreted as a projection of the corresponding signal 5(r] )S (r2) onto the subdiagonal in the space defined by r and rz. 

The average propagator is introduced to describe the echo attenuation in the displacement experiments of Fig. 5.4.4 in terms of the particle displacement R along the gradient direction during the gradient-pulse delay A. To this end (5.4.26) is transformed to the rotated coordinate frame of Fig. 5.4.5 with axes defined in (5.4.21). By the same derivation used for (5.4.22) one arrives at... 

In the long-time limit A d /D the propagator has a simple relationship to the geometry of closed pores. In this case, the conditional probability density P(rj r2. A) in (5.4.27) is independent of the starting position r so that it reduces to the spin density Mo(r2) = Mo(ri -t- R) of the pore. In consequence, the average propagator (5.4.30) becomes an auto-correlation function of the spin density (cf. eqn (5.4.22)),... 

CLSM micrograph of a heterogeneous emulsion that is used directly in the simulations of the average propagator, (a) The bright phase is the fat phase (b) The gray phase is the water phase. 

It is useful to define the average propagator, Pj(R, A) as the ensemble-averaged probability for a molecule to displace by R = r — r irrespective of starting position.Thus,... 

Note that the spin-echo signal intensity will also be affected by NMR relaxation during the pulse sequence. However, the time taken for echo formation may be kept constant while varying q, and this enables the average propagator to be determined through the inverse Fourier relationship ... 

In an isolated pore, collisions of molecules with the pore wall during the time A mean that the apparent diffusion rate will be influenced by the pore geometry. The average propagator will still be Gaussian if the distance moved by molecules in the time A is less than the pore size (i.e. when Aspherical pore), and the measured diffusion coefficient will then be the same as for the case of unrestricted diffusion (i.e. Eqs (25) and (26) can be applied). The pore wall will, however, have an effect if A>r /D. In the case of isolated pores, the pore walls impose a limit to the distance that molecules can diffuse. Figure 7 shows a schematic plot of ln( (, A)) against for the cases of... 

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