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. By neglecting interactions among monomers, this model assumes that two (or more) monomers can occupy the same location. Although it is simple, its generality gives insight about the physics of polymers. In this model, monomers are rigid rods of a fixed length l, and their orientation is completely independent of the orientations and positions of neighbouring monomers. In some cases, the monomer has a physical interpretation, such as an amino acid in a polypeptide. In other cases, a monomer is simply a segment of the polymer that can be modeled as behaving as a discrete, freely jointed unit. If so, l is the Kuhn length. For example, chromatin is modeled as a polymer in which each monomer is a segment approximately 14–46 kbp in length.
Model N mers form the polymer, whose total unfolded length is: L = N l , {\displaystyle L=N\,l,} where N is the number of mers, and l is the length of each mer. In this very simple approach where no interactions between mers are considered, the energy of the polymer is taken to be independent of its shape, which means that at thermodynamic equilibrium, all of its shape configurations are equally likely to occur as the polymer fluctuates in time, according to the Maxwell–Boltzmann distribution. Let us call R → {\displaystyle {\vec {R}}} the total end to end vector of an ideal chain and r → 1 , … , r → N {\displaystyle {\vec {r}}_{1},\ldots ,{\vec {r}}_{N}} the vectors corresponding to individual mers. Those random vectors have components in the three directions of space. Most of the expressions given in this article assume that the number of mers N is large, so that the central limit theorem applies. The figure below shows a sketch of a (short) ideal chain.
The two ends of the chain are not coincident, but they fluctuate around each other, so that of course:
⟨ R → ⟩ = ∑ i = 1 N ⟨ r → i ⟩ = 0 → {\displaystyle \left\langle {\vec {R}}\right\rangle =\sum _{i=1}^{N}\left\langle {\vec {r}}_{i}\right\rangle ={\vec {0}}~}
Throughout the article the ⟨ ⟩ {\displaystyle \langle \rangle } brackets will be used to denote the mean (of values taken over time) of a random variable or a random vector, as above. Since r → 1 , … , r → N {\displaystyle {\vec {r}}_{1},\ldots ,{\vec {r}}_{N}} are independent, it follows from the central limit theorem that R → {\displaystyle {\vec {R}}} is distributed according to a normal distribution (or gaussian distribution): precisely, in 3D, R x , R y , {\displaystyle R_{x},R_{y},} and R z {\displaystyle R_{z}} are distributed according to a normal distribution of mean 0 and of variance: σ 2 = ⟨ R x 2 ⟩ − ⟨ R x ⟩ 2 = ⟨ R x 2 ⟩ − 0 {\displaystyle \sigma ^{2}=\langle R_{x}^{2}\rangle -\langle R_{x}\rangle ^{2}=\langle R_{x}^{2}\rangle -0}
⟨ R x 2 ⟩ = ⟨ R y 2 ⟩ = ⟨ R z 2 ⟩ = N l 2 3 {\displaystyle \left\langle R_{x}^{2}\right\rangle =\left\langle R_{y}^{2}\right\rangle =\left\langle R_{z}^{2}\right\rangle =N\,{\frac {l^{2}}{3}}}
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