Vector-averaging model

Diagram A shows predictions made by Cheng (1994) about where pigeons should search when landmark vectors indicated the location of food in two different places. Diagram B shows that pigeons in an experiment searched along the arc of the circle predicted by the direction averaging model. 

Here the vector rj represents the centre of mass position, and D is usually averaged over several time origins to to improve statistics. Values for D can be resolved parallel and perpendicular to the director to give two components (D//, Dj ), and actual values are summarised for a range of studies in Table 3 of [45]. Most studies have found diffusion coefficients in the 10 m s range with the ratio D///Dj between 1.59 and 3.73 for calamitic liquid crystals. Yakovenko and co-workers have carried out a detailed study of the reorientational motion in the molecule PCH5 [101]. Their results show that conformational molecular flexibility plays an important role in the dynamics of the molecule. They also show that cage models can be used to fit the reorientational correlation functions of the molecule. 

The instantaneous composition of a copolymer X formed at a monomer mixture composition x coincides, provided the ideal model is applicable, with stationary vector ji of matrix Q with the elements (8). The mathematical apparatus of the theory of Markov chains permits immediately one to wright out of the expression for the probability of any sequence P Uk in macromolecules formed at given x. This provides an exhaustive solution to the problem of sequence distribution for copolymers synthesized at initial conversions p l when the monomer mixture composition x has had no time to deviate noticeably from its initial value x°. As for the high-conversion copolymerization products they evidently represent a mixture of Markovian copolymers prepared at different times, i.e. under different concentrations of monomers in the reaction system. Consequently, in order to calculate the probability of a certain sequence Uk, it is necessary to average its instantaneous value P Uk over all conversions p preceding the conversion p up to which the synthesis was conducted. 

Fig. 34. Snapshots of the bubble hold-up in the DBM model without bubble coalescence, and the time average vector plot of the emulsion phase after 100s of simulation (a) + (b) 0 = 0.1.

Even better agreement is observed between calorimetric and elastic Debye temperatures. The Debye temperature is based on a continuum model for long wavelengths, and hence the discrete nature of the atoms is neglected. The wave velocity is constant and the Debye temperature can be expressed through the average speed of sound in longitudinal and transverse directions (parallel and normal to the wave vector). Calorimetric and elastic Debye temperatures are compared in Table 8.3 for some selected elements and compounds. 

Because two arbitrary bond vectors are uncorrelated in this simple model, the thermal average over the scalar product of two different bond vectors vanishes, (r/ rjt) = 0 for j k, while the mean squared bond vector length is simply given by (i ) = a. It follows that the mean squared end-to-end radius Rjf is proportional to the number of monomers. 

AMJhj is a function 6f a) the orientation of the inter-crosslink vector h. with the respect to the magnetic field B0 and b) the time-averaged value of the sum over k, which is the actual measure of the motional restrictions induced by crosslinking. In lightly crosslinked networks, presented by the freely-jointed model of the polymer chain 108), the residual part also can be described by the number of statistical segments in the chain section between crosslinks (Z) (Eq. (24)) ... 

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