Space pseudophase

Figure 3.5 The Rossler strange attractor. (A) The phase space. (B) The state variable yi (t). (C) Reconstruction in the pseudophase space.

Since (11.12) has an infinite number of degrees of freedom [523], we constructed a pseudophase space [4,32] for the system of (11.12) and (11.13) using the model variables c(t), c(t + t°/2), c(t + t°), Figure 11.11. The use of three dimensions is in accordance with the embedding dimension that Ilias et al. [514] have found. The attractor of our system is quite complicated geometrically, i.e., it is a strange attractor. The real phase space is of infinite dimension. However, trajectories may be considered to lie in a low-dimensional space (attractor). The model parameters take the same values as in Figure 11.10 and time runs for 10 days. 

Cover illustration Left panel Stochastic description of the kinetics of a population of particles, Fig 9.15. Middle panel Dissolution in topologically restricted media, Fig. 6.8B (reprinted with permission from Springer). Right panel A pseudophase space for a chaotic model of cortisol kinetics, Fig.11.11. 

Since we are interested in classifying conformational pseudophases of polymers with respect to their thickness, it is useful to introduce the restricted conformational space 7 = X rge(X) > p of all conformations X with a global radius of curvature larger... 

At very low temperatures, i.e., in pseudophase ACl, we have argued in the previous section that the dominant polymer conformation is the most compact single-layer film. This is confirmed by the behavior of R and Rj ), the latter being zero in this phase. A simple argument that the structure is indeed maximally compact is as follows. It is well known that the most compact shape in the two-dimensional continuous space is the circle. For n monomers residing in it, n nr, where r is the (dimensionless) radius of this circle. The usual squared gyration radius is... 

The first important observation is that the diagram is divided into two separate regions, the pseudophases of desorbed conformations (DC and DE) and the remaining different phases of adsorption. The space in-between is blank, i.e., none of these (possible) conformations represents a free-energy minimum state. This shows that transitions between the adsorbed and desorbed pseudophases are always first-order-like. It should be noted that the regime of contact pairs lying above the shown compact phases is forbidden,... 

Eventually, let us compare the adsorption behavior with what we had found in Chapter 13 for simplified hybrid lattice models of polymers and peptides near attractive substrates. The adhesion of the jjeptides at the Si(lOO) substrate exhibits very similar features. Exemplified for peptide S3, Fig. 14.13 shows the plot of the canonical probability distributionpcan E, q) 8 E — E(X))S(q — q(X))) at room temperature (T = 300 K). The peak at E, q) (80.5 kcal/mole, 0.0) corresponds to conformations that are not in contact with the substrate. It is separated from another peak near (E, q) (74.5 kcal/mole, 0.2) and belongs to conformations with about 17% of the heavy atoms with distances < 5 A from the substrate surface (compare with Fig. 14.12). That means adsorbed and desorbed conformations coexist and the gap in between the peaks separates the two pseudophases in g -space, which causes a kinetic free-energy barrier. Thus, the adsorption transition is a first-order-like pseudophase transition in q, but since both structural phases (adsorbed and desorbed) coexist almost at the same energy, the transition in E space is weakly of first order. ... 

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