Random walk statistics

Next let us apply random walk statistics to three-dimensional chains. We begin by assuming isolated polymer molecules which consist of perfectly flexible chains. 

When we discussed random walk statistics in Chap. 1, we used n to represent the number of steps in the process and then identified this quantity as the number of repeat units in the polymer chain. We continue to reserve n as the symbol for the degree of polymerization, so the number of diffusion steps is represented by V in this section. 

Figure S5.5 Random-walk statistics Each plot shows the number of atoms, N, reaching a distance d in a random walk, for walks (a) 100 atoms and 200 steps, (b) 500 atoms and 200 steps, and (c) 10,000 atoms and 400 steps. The curve approximates to the binomial distribution as the number of atoms and steps increases.

In addition, the random-walk statistics show that... 

Conformation and Deformation of Linear Macromolecules in Concentrated Solutions and Melts in the Self-Avoiding Random Walks Statistics... 

Self-avoiding random walks statistics for intertwining polymeric chains and based on it thermodynamics of their conformational state in m-ball permitted to obtain the theoretical expressions for elasticity modules and main tensions appearing at the equilibrium deformation of /n-ball. Calculations on the basis of these theoretical expressions without empirical adjusting parameters are in good agreement with the experimental data. 

In presented work the analysis of osmotic pressure of the polymeric solutions has been done with taken into account the thermodynamics of conformation state of macromolecules following from the self-avoiding random walks statistics [13, 14],... 

Medvedevskikh Yu. G. Conformation and deformation of linear macromolecules in concentrated solutions and melts in the self-avoiding random walks statistics (see paper in presented book)... 

The statistical basis of diffusion requires arguments that may be familiar from kinetic molecular theory. Elementary concepts from the theory of random walks and its relation to diffusion form the third topic, which is covered in Section 2.6. As is well known, the random walk statistics can also be used for describing configurational statistics of macromolecules under some simplifying assumptions this is outlined in Section 2.7. 

In this section we review random walk statistics and their relation to diffusion. More elaborate discussions of these and related topics are available in Berg (1993) the collection of the original papers of Einstein (reprinted in Einstein 1956) is another excellent source of this material. 

The dimensions of a randomly coiled polymer molecule are a topic that appears to bear no relationship to diffusion however, both the coil dimensions and diffusion can be analyzed in terms of random walk statistics. Therefore we may take advantage of the statistical argument we have developed to consider this problem. 

Describe how the random walk statistics are used to relate the random walk to the diffusion coefficient. 

What assumptions are made in the development of the random walk statistics discussed in the text, and what do they imply for the correspondence between random walk and diffusion ... 

Why can one use random walk statistics to derive expressions for the end-to-end distance of a polymer chain Under what conditions can this be done ... 

Berg, H. C., Random Walks in Biology, expanded ed. Princeton University Press, Princeton, NJ, 1993. (Undergraduate level. A lucid introduction to random walk statistics. Discusses translational and rotational diffusion, self-propelled motion, random walk, and sedimentation Relevant to biological species of colloidal dimensions, but no background in biology is needed.)... 

Unfortunately, for the investigation of random walk statistics in the regular 3D lattice of obstacles the approach based on the idea of conformal transformations cannot be applied. Nevertheless, due to the analogy established in the 2D-case it is naturally to suppose that between random paths statistics in the 3D lattice of uncrossable strings and the free random walk in Lobachevsky space the similar analogy remains. Let us present below some arguments confirming that idea. 

Gaussian coils are characterized by a gaussian probability distribution [2] for the monomers and describe adequately flexible polymer blocks. Ideal chains follow random walk statistics, i.e.,... 

Within Flory s mean field approximation, biased random walk statistics are characterized by ... 

Random-Walk Statistics The Freely Jointed Chain... 

We briefly comment on some other treatments. One of the oldest precursors comes from Singer ), who applied lattice theory but assumed all segments to be restricted to the train layer. Frisch and Slmha ) presented a model accounting for loops and tails in addition to trains, using random-walk statistics with a Boltzmann factor for train segments. However, their statistical treatment is incorrect... 

Every possible conformation of an ideal chain can be mapped onto a random walk. A particle making random steps defines a random walk. If the length of each step is constant and the direction of each step is independent of all previous steps, the trajectory of this random walk is one conforma-tion of a freely jointed chain. Hence, random walk statistics and ideal chain statistics are similar. 

Consider a monomer of an ideal polymer trying to reach fellow monomers of the same chain via a CB radio (see Fig. 2.17). The number of monomers it can call depends on the range r of its transmitter. It can contact any monomer within the sphere of radius r of itself The number of monomers m that can be reached via a CB radio with range r is given by random walk statistics ... 

The previous problem showed that the equivalent freely jointed chain follows random walk statistics even if the effective monomer is renormalized to be larger than b. What is the smallest effective monomer size for which this renormalization works ... 

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