Editing Random coil (section)

==Random walk model: The Gaussian chain==
{{main|Ideal chain}}
[[Image:Ideal chain random walk.svg|thumb|200px|Short [[ideal chain|random chain]]]]

There are an enormous number of different [[ Ludwig Boltzmann#Physics |ways]] in which a chain can be curled around in a relatively compact shape, like an unraveling ball of twine with much open [[space]], and comparatively few ways it can be more or less stretched out. So, if each conformation has an equal [[probability]] or [[statistics|statistical]] weight, chains are much more likely to be ball-like than they are to be extended — a purely [[entropy|entropic]] effect. In an [[statistical ensemble (mathematical physics)|ensemble]] of chains, most of them will, therefore, be loosely [[sphere|balled up]]. This is the kind of shape any one of them will have most of the time.

Consider a linear polymer to be a freely-jointed chain with ''N'' subunits, each of length <math>\scriptstyle\ell</math>, that occupy [[0 (number)|zero]] [[volume]], so that no part of the chain excludes another from any location. One can regard the segments of each such chain in an ensemble as performing a [[random walk]] (or "random flight") in three [[dimension]]s, limited only by the constraint that each segment must be joined to its neighbors. This is the ''[[ideal chain]]'' [[mathematical model]]. It is clear that the maximum, fully extended length ''L'' of the chain is <math>\scriptstyle N\,\times\,\ell</math>. If we assume that each possible chain conformation has an equal statistical weight, it can be [[ideal chain|shown]] that the probability ''P''(''r'') of a polymer chain in the [[statistical population|population]] to have distance ''r'' between the ends will obey a characteristic [[Probability distribution|distribution]] described by the formula

: <math>P(r) = 4 \pi r^2 \left(\frac{3}{2\; \pi \langle r^2\rangle}\right)^{3/2} \;e^{-\,\frac{3r^2}{2\langle r^2\rangle}}</math>

where <math>{\langle r^2\rangle}</math> is the [[mean]] of <math>{r^2}</math>.

The ''average'' ([[root mean square]]) end-to-end distance for the chain, <math>\scriptstyle \sqrt{\langle r^2\rangle}</math>, turns out to be <math>\scriptstyle\ell</math> times the square root of&nbsp;''N'' &mdash; in other words, the average distance scales with ''N'' <sup>0.5</sup>.