Freely Jointed Chain — Polymer Conformations

A polymer of N segments with bond length b performs a random walk. The end-to-end distance follows ⟨R²⟩ = Nb², with a Gaussian distribution for large N (Central Limit Theorem).

Chain length N 50

Bond length b 1.0

Excluded volume ν 0.00 ν>0: self-avoidance (SAW). ν=0.588 in 3D (Flory).

Stretching force f·b/k_BT 0.00

Samples per frame 20

⟨R²⟩ theory: —
⟨R²⟩ measured: —
R_max = Nb: —
Samples: 0

Flory (1949): ⟨R²⟩ = Nb² (ideal)
SAW: ⟨R²⟩ ~ N^{2ν} b² with ν≈0.588 (3D), ν=3/4 (2D)

Force-extension (Langevin):
⟨z⟩/Nb = L(fb/k_BT)
L(x) = coth(x) − 1/x

WLC (worm-like chain): ⟨z⟩ = Lc[1 − (1/4)(k_BT/fb)^{1/2}]

Single chain conformation (random walk in 2D projection)

Ensemble: end-to-end vector cloud (N=50). Red = Gaussian fit.