Entanglement macroscopic

The two main amphibole asbestos fibers are amosite and crocidoHte, and both are hydrated siHcates of iron, magnesium, and sodium. The appearance of these fibers and of the corresponding nonfibrous amphiboles is shown in Figure 1. Although the macroscopic visual aspect of clusters of various types of asbestos fibers is similar, significant differences between chrysotile and amphiboles appear at the microscopic level. Under the electron microscope, chrysotile fibers are seen as clusters of fibrils, often entangled, suggesting loosely bonded, flexible fibrils (Fig. 2a). Amphibole fibers, on the other hand, usually appear as individual needles with a crystalline aspect (Fig. 2b). 

An even deeper dilemma confronts an honest physicist when trying to understand what really happens during a quantum measurement. Despite using (macroscopically) intuitive descriptions of the quantum-entangled objects, what really happens remains a deep mystery. Or, consider the following simple thought experiment using CA. 

In summary, the chain dynamics for short times, where entanglement effects do not yet play a role, are excellently described by the picture of Langevin dynamics with entropic restoring forces. The Rouse model quantitatively describes (1) the Q-dependence of the characteristic relaxation rate, (2) the spectral form of both the self- and the pair correlation, and (3) it establishes the correct relation to the macroscopic viscosity. 

So far, the existence of a well-defined entanglement length in dense polymer systems has been inferred indirectly from macroscopic experiments like measurements of the plateau modulus. However, its direct microscopic observation remained impossible. The difficulty in directly evaluating the entanglement... 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

In the case of filled systems, the two latter effects provide a substantial contribution to C2 compared with the influence of trapped entanglements [80]. For filled systems, the estimated or apparent crosslinking density can be analyzed with the help of the Mooney-Rivlin equation using the assumption that the hard filler particles do not undergo deformation. This means that the macroscopic strain is lower than the intrinsic strain (local elongation of the polymer matrix). Thus, in the presence of hard particles, the macroscopic strain is usually replaced by a true intrinsic strain ... 

Spring-bead models relate frictional force to the relative velocity of the medium at the point of interaction. The entanglement friction coefficient above is defined in terms of the relative velocity of the passing chain. Since the coupling point lies, on the average, midway between the centers of the two molecules involved, the macroscopic shear rate must be doubled when applying the result to a spring-bead model. Substitution of 2 CE for Con in the Rouse expression for viscosity yields... 

It was also assumed that internal times such as xe are proportional to the macroscopic viscosity xe = t]/ri0 (xe)0 (xe)0 or x0 is the mean entanglement time at equilibrium. The viscosity was then calculated from the expression tj = v/y2,... 

The point where these different strands meet is as follows. In one dynamical network approach, of Manfred [73] macroscopic spacetime is considered to be a coarse-grained emergent phenomenon (called an order parameter manifold). It is assumed to be the result of some kind of geometric phase transition (very much in the spirit of the physics of self-organization). This framework was developed in quite some detail [75]. We argued in Ref. 76 that what we consider to be the elementary building blocks of continuous spacetime—the so-called physical points—are, on a finer scale, actually densely entangled subclusters of... 

---