Gaussian Chain Field Theory

End-to-end distribution · structure factor · Rouse modes · relaxation spectrum

Chain length N 100 Kuhn length b 1.0 Excluded vol. v 0.0 Rouse modes shown 5

⟨R²⟩ = Nb²: —

Rg² = Nb²/6: —

τ_Rouse ∝ N²: —

Flory ν: —

Gaussian chain: each bond vector is drawn from P(b) ∝ exp(−3b²/2b₀²). The end-to-end distribution is Gaussian: G(R,N) = (3/2πNb²)^(3/2) exp(−3R²/2Nb²). The structure factor S(q) = (2/x²)[exp(−x)−1+x], x=q²Rg², is the Debye function. Rouse modes X_p = (1/N)∫₀ᴺ R(s)cos(pπs/N)ds diagonalize the chain dynamics; relaxation times τ_p = τ₁/p² give the Rouse spectrum. Excluded volume (v>0) swells the chain with Flory exponent ν≈0.588 (3D), changing R ~ N^ν.