Trajectory, blob

Here Ry stands for a typical ionization potential. Equation (2) indicates that the terminal positron blob is a spherical nanovolume, which confines the end part of its trajectory. This is where ionization slowing down is the most efficient (the thermalization stage of the subionizing positron is not included here). The mathematical formulation of this statement is twofold. Just after the first blob formation step , which is ltr Wu) (the thick arrow in Fig. 5.1), the positron reaches the center of the blob. After that, the end part of the ionization slowing-down trajectory is embraced by the blob i.e., the slowing-down displacement of the positron, Rion Wbi, Ry) — au is equal to the radius of the blob, au-... 

The physical meaning of a tension blob is the length scale at which external tension changes the chain conformation from almost undeformed on length scales smaller than to extended on length scales larger than The trajectory of the stretched chain (Fig. 2.13) shows that each tension... 

Hence, if we start with a enormous solid blob of initial conditions, it eventually shrinks to a limiting set of zero volume, like a balloon with the air being sucked out of it. All trajectories starting in the blob end up somewhere in this limiting set later we ll see it consists of fixed points, limit cycles, or for some parameter values, a strange attractor. 

The motion on the attractor exhibits sensitive dependence on initial conditions. This means that two trajectories starting very close together will rapidly diverge from each other, and thereafter have totally different futures. Color Plate 2 vividly illustrates this divergence by plotting the evolution of a small red blob of 10,000 nearby initial conditions. The blob eventually spreads over the whole attractor. Hence nearby trajectories can end up anywhere on the attractor The practical implication is that long-term prediction becomes impossible in a system like this, where small uncertainties are amplified enormously fast. 

This integral simply counts the number of blobs along the trajectory of the chain. For flat layers this reduces to the familiar Ff.hainl N a /, For spherical layer Fchain/ F + F)/Rin- Extended cylindrical brushes, L > iij ,... 

An alternative derivation of Eq. (B-9) is based on the decimation procedure. Rigour-ously, a detailed analysis in terms of renormalization group trajectories (cf. Ref. 22, Chap. 11 and Ref. 37) is required. This type of procedure is the theoretical basis behind the so-called blob model. The excluded volume effects are important at short-range scale, within a blob containing g monomers. At larger scale excluded volume interactions are screened. The mean-field approach will, therefore, be valid if the blob of size and volume is taken as the site. Then, in order to describe the thermodynamics, the following transformations must be carried out in Eq. (B-6)... 

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