Polymer as Random Walk — End-to-End Distance

A polymer chain is a random walk — end-to-end distance scales as N^ν where ν≈3/5 in 3D (Flory)

Chain length N: 60 Ensemble size: 30

Current R (SAW): — | Current R (ideal): — | Fitted ν (SAW): —

Self-avoiding (SAW) Ideal random walk Flory ν=0.588

Flory (1949): A polymer in good solvent is a self-avoiding walk. Excluded volume swells the chain: R ~ N^ν with ν=3/(d+2) (Flory) — exactly 3/4 in 2D, ~3/5 in 3D (exact: 0.5877...). Ideal Gaussian chain has ν=1/2. SAW is one of the hardest problems in combinatorics — exact enumeration is exponentially hard.