Taylor expansions for random-walk polymers

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Original entry by Joerg Fritz, AP225 Fall 2009

Keywords

Random walk, Taylor expansion, Symmetry, Dimensionality

Outline

This entry investigates a small aspect of the very elegant derivation for the scaling of end-to-end distance as described in detail here. We specifically ask the following question: Can we predict the probability <math>p(n,\vec{r}-\vec{r_1})</math> that a polymer with n segments has a vector <math>\vec{r_1}</math> connecting its two ends, if <math>\vec{r_1}</math> is close to <math>\vec{r}</math> and we know the probability for <math>\vec{r}</math>, that is <math>p(n,\vec{r})</math> and its spacial derivatives? Even more specifically, if <math>p(n,\vec{r})</math>

Taylor expansion in several variables

Fig.2 Geometric representation of the flat sheet shown in figure 1, the numerical discretization and the result of the simulation in case of successful folding.

Polymers in lower dimensions

An intuitive explanation

The mathematical way to explain it