Self-intersecting walks

FIGURE 5.4. Two dimensional illustration of the MC growth of non-self-intersecting walks (see the text for details). On the left side (A) an example of the successful structure composed of n = 15 segments is shown. On the right side (B) an intersection has been detected before reaching n = 15, the final chain length, and the chain has to be removed from the statistical ensemble. 

NON-SELF-INTERSECTING RANDOM WALKS IN LATTICES WITH NEAREST-NEIGHBOR INTERACTIONS... 

Interactions, Near-Neighbor, Non-Self Intersecting Random Walks... 

Non-Self Intersecting Random Walks on Lattices with Near-Neighbor... 

The above calculations assume that the gross chain conformations are those of a random walk, which is the case in the melt. However, for an isolated polymer molecule in a dilute solution, the average conformation is affected by excluded-volume interactions between one part of the chain and another. Because the chain must avoid self-intersection, the conformation of the chain will be that of a self-avoiding walk, rather than a random walk, if the solution is athermal—that is, if all interactions are negligible except excluded volume. Self-avoiding walks lead, on average, to more expanded coil dimensions, since expanded configurations are less likely than contracted ones to lead to self-intersection of the chain. Thus, in an athermal solution, the mean-square end-to-end dimension of a polymer molecule scales as... 

Estimate the A-dependence of the success rate of a simple Monte Carlo simulation of a self-avoiding walk with 100 steps. Assume that random walks are generated on a cubic lattice. Each step of the walk is not allowed to step back (they can only go forward, up, down, turn left, or right with equal probability of 1/5). Walks that intersect themselves are discarded. 

Figure 4 Chains of N = 11 steps (bonds), that is, 12 monomers (solid circles) on a square lattice (open circles). Immediate chain reversals are not allowed therefore the maximum directions v available are 4 for the first step and 3 for the later steps, (a) An ideal chain (random walk) starting from the origin (1) the chain intersects itself and the last step (dashed line) goes on the third one. (b) A self-avoiding walk (SAW) is not allowed to self-intersect. The SAWs are a subgroup of the ideal chains, (c) A SAW with a finite interaction e between nonbonded nearest-neighbor monomers. The total energy of the chain is 6e.

The condition of the absence of self-intersection trajectories of walks requires from the point of view of the links of star per cells distribution that in every cell of the lattice space should be exist not more than one link of a star. The links of a polymeric chain are continuous they cannot be separated one from other and located upon the cells into the undefined order. Therefore, the number of a links in a chain is its essential distinctive feature. The links of the different rays are also distinguishable That is why the number of different methods location of sN distinguishable links of a star per Z similar cells under condition that in every cell carmot be more... 

The first term is positive and is more than the second one it takes into account the all trajectories of walk with imposed on them singular limitation of the connectedness of the links into a chain, and doesn t accept the reverse step. The second term is negative (co 2N) < 1) it takes into account the additional limitations on the trajectories of walk by requirement of their self-intersection absence. At this, the first term at given data s, N, d is... 

The study of pol5rmeric fractals in different states started directly after the publication of Mandelbrot s fractal conception [1], that is, about at the end of 70s of the last century. The cause of the indicated approach application for a pol5rmer macromolecules analysis is obvious enough it is difficult to find a more suitable object for fiuctal models application. One from the first research in this field with the experimental method of electrons spin-relaxation for proteins [2] found, out that the protein macromolecule dimension was equal to 1.65 0.04, that coincides practically exactly with the dimension of 5/3 of random walk without self-intersections, by which a polymeric macromolecule is usually simulated [3]. 

The authors [4, 5] considered random walks without self-intersections or self-avoiding walks (SAW) as the critical phenomenon. Monte-Carlo s method within the framework of computer simulation confirmed SAW or polymer chain self-similarity, which is an obligatory condition of its fac-tuality. Besides, in Ref [4], the main relationship for polymeric fractals at their treatment within the framework of Flory conception was obtained ... 

In the context of a simple lattice model the problem could be formulated as follows. Compute the number of non-self-intersecting random walks on the lattice and the distribution of the segment density, size, shape, etc., of the resulting random coils as a function of the chain length. The first thought is to use computer for an exact enumeration of all possible conformations of a n-segment chain. Unfortu-... 

It has been noticed several times [3, 4] that the configuration of soft polymer chains differs from the RW trajectory in one important aspect it must not intersect. This limitation, known as long-range ordering effect or excluded volume effect, requires new statistics, i.e., statistics of self-avoiding walks (SAW). The attempts made so far [3] have not succeeded in solving this problem completely. 

We see that the direct data on coils are not quite conclusive. It is then helpful to return to theoretical calculations. There do exist rather accurate numerical studies on real chains on a lattice. The chain is still represented by a random walk as in Fig. (1.1), but the main difference is that now this walk can never intersect itself. We call it a self-avoiding walk (SAW). 

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