Random walk chain

Vincent analyzed the tensile fracture stress o, of a broad range of polymers as a function of the number of backbone bonds per cross sectional area ( 2) and found a nearly linear relation, o 2, as shown in Fig. 12. 2 is related to via the theory of entanglements for random walk chains as [74]... 

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

Note 1 For models in which the segments are not all uniform in length, the name random-walk chain has been used. 

Fig. 2.1. A random walk chain of 50 monomers of fixed length (thin polygon), leaking the position of every fifth monomer as segment coordinates, we find our model chain (fal polygon), which is approximated by a Gaussian coil...

In the following two sections we first consider colls In solutions which are so dilute that coll-coll Interaction does not play a role. Section 5.2a deals with random-walk chains where the Interaction between the units within one chain may be neglected. We denote these as ideal chains. Section 5.2b treats swollen coils In which the monomeric units repel each other due to so-called excluded volume effects. 

Start with Eq. (2-14a), the definition of R, . For a random-walk chain, each step of the walk is independent of the others. Hence, the ensemble average can be brought inside the summations. Thus,... 

Consider a random-walk chain of no monomer units in a n dium which is densely filled with the contours of other chains. For the moment take the ends of the test chain to be fixed. Let its surrounding be r resented by a permanently connected rigid lattice of uncrossable lines enveloping the chain contour. We assume that the effect of this obstade lattice on the conformations of the chain is specified simply by a distance scale, the mesh so d, as follows. Pieces of the diain which have a mean quare end-to-end distance (r ) much smaller than (F can explore aU conformations with the same probability as free chains of the same length. For loiter pieces, the presence of the obstacles (and the fact that the pieces are connected in a definite sequence between the fixed end points of the... 

FIGURE 10.1 (a) Random walk chain of 32 steps, length / and (b) cosine law for two bonds. 

Naturally, chains in solvents close to the theta state must become very long in order to attain the asymptotic self-avoiding behavior. The chain size required for self-avoidance to become significant is called the thermal blob length [8.3]. Chains smaller than a thermal blob size are approximately random-walk chains. Even chains much larger than this size behave as simple random walk fractals with D = 2 on length scales r < ft- This goes to infinity as theta-solvent conditions are approached. 

Fig. 2.8 Sketch of a random walk chain consisting of monomers with length b. For any given walk the end-to-end vector R = 5)) r,...

Figure 32.3 Example of a random-walk chain conformation. Source LRC Treloar, Physics of Rubber Elasticity, 2nd edition, Oxford University Press, New York, 1958.

Note we want to reach the long-run distribution as quickly as possible. There is a trade off between the number of candidates accepted and the distance moved. Generally a chain with a random-walk candidate density will have more candidates accepted, but each move will be a shorter distance. The chain with an independent candidate density will have fewer moves accepted, but the individual moves can be very large. We can see this in comparing the traceplot of the random-walk chain in Figure 6.2 with the trace plot of the independent chain Figure 6.5. The random-walk... 

Since this is a random-walk chain, we we will always accept a candidate value that is uphill from the current value. A candidate value that is downhill from the current value will be accepted with probability a = Wfe start the chain at initial... 

When the Metropolis-Hastings algorithm is run blockwise and the parameters in different blocks are highly correlated, the candidate for a block won t be very far from the current value for that block. Because of this, the chain will move slowly around the parameter space very slowly. We observed this in Figure 6.15 and in the traceplots in Figure 6.16. The traceplots of the parameters of a blockwise Metropolis-Hastings chain look much more like those for a random-walk chain than those for an independent chain. 

Figure 7.5 The sample autocorrelation function for the random-walk chain and the independent chain, respectively.

When the parameters are strongly correlated, a blockwise Metropolis-Hastings chain will move slowly through the parameter space. The trace plots will look similar to those from a random-walk chain. 

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