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In polymer physics, random walk describes an ideal chain. It is the simplest model to study polymers. [36] In other fields of mathematics, random walk is used to calculate solutions to Laplace's equation, to estimate the harmonic measure, and for various constructions in analysis and combinatorics.
The simplest formulation of excluded volume is the self-avoiding random walk, a random walk that cannot repeat its previous path. A path of this walk of N steps in three dimensions represents a conformation of a polymer with excluded volume interaction. Because of the self-avoiding nature of this model, the number of possible conformations is ...
An ideal chain (or freely-jointed chain) is the simplest model in polymer chemistry to describe polymers, such as nucleic acids and proteins.It assumes that the monomers in a polymer are located at the steps of a hypothetical random walker that does not remember its previous steps.
Also, many polymers have random branching. Even with corrections for local constraints, the random walk model ignores steric interference between chains, and between distal parts of the same chain. A chain often cannot move from a given conformation to a closely related one by a small displacement because one part of it would have to pass ...
Since the space occupied by a segment in the polymer chain cannot be taken by another segment, a self-avoiding random walk model can also be used. The Kuhn segment construction is useful in that it allows complicated polymers to be treated with simplified models as either a random walk or a self-avoiding walk , which can simplify the treatment ...
The polymer chain will behave exactly as predicted by the random walk or ideal chain model. This makes experimental determination of important quantities such as the root mean square end-to-end distance or the radius of gyration much simpler. Additionally, the theta condition is also satisfied in the bulk amorphous polymer phase. Thus, the ...
Short random chain. One extremely naive yet fruitful approach to quantitatively analyze the spatial structure and configuration of a polymer is the free random walk model. . The polymer is depicted as a chain of point like unit molecules which are strongly bound by chemical bonds and hence the mutual distance between successive units can be approximated to be const
It is the most common measure of the spatial extent of random motion, and can be thought of as measuring the portion of the system "explored" by the random walker. In the realm of biophysics and environmental engineering , the Mean Squared Displacement is measured over time to determine if a particle is spreading slowly due to diffusion , or if ...