= Solution
\b[Ideal and self-avoiding walks.] An <ideal chain> has independent isotropic coarse-grained steps and permits self-intersections in its mathematical model. For $N$ steps of length $b$, the cross terms average to zero, so its root-mean-square <end-to-end distance of a polymer chain> obeys
$$
\boxed{\langle R^2\rangle=Nb^2,\qquad R_{\mathrm{rms}}\propto N^{1/2}}.
$$
A <self-avoiding walk> excludes repeated occupancy or intersection at the segment scale. This represents positive <excluded volume> in a dilute <good solvent>. The nonlocal constraint correlates otherwise distant portions of the chain and swells it:
$$
\boxed{R_{\mathrm{rms}}\propto bN^\nu,\qquad \nu\simeq0.588\quad\text{in three dimensions}}.
$$
The <Flory approximation> replaces this exponent by $3/5$, by balancing stretching free energy $R^2/(Nb^2)$ with an interaction contribution $vN^2/R^3$. The result is a more extended and less internally dense coil than the ideal walk; self-avoidance changes the statistics, not merely the local step length.
\b[Two ideal-chain regimes.] At the <theta temperature> for a dilute polymer–solvent pair, leading effective segment attractions and repulsions cancel. Large-scale chain statistics are approximately ideal, even though microscopic atoms still have finite size. In a homogeneous <polymer melt>, surrounding chains screen long-range <excluded volume> interactions of a chosen chain, again making its equilibrium large-scale size approximately ideal. Local bond-angle correlations must be included in an effective statistical length such as the <Kuhn length>. <Polymer entanglements> can slow the dynamics without invalidating this leading equilibrium size law.
\b[Semiflexible DNA.] The <nucleic acid double helix> gives double-stranded <DNA> a bending stiffness and substantial local orientational persistence. Independent step orientations are therefore a poor description on short scales, while self-avoidance alone does not specify the energy cost of bending. The <worm-like chain> describes the molecule as a continuously bendable, locally inextensible curve $\mathbf r(s)$ parameterized by <arc length> $s$, with unit tangent $\mathbf t(s)=d\mathbf r/ds$. Its bending energy is
$$
E_b=\frac A2\int_0^L\left|\frac{d\mathbf t}{ds}\right|^2ds,\qquad |\mathbf t|=1,
$$
where $A$ is <bending stiffness>. In the free three-dimensional model, the <persistence length> is $\ell_p=A/(k_BT)$ and
$$
\boxed{\langle\mathbf t(s)\cdot\mathbf t(s+u)\rangle=e^{-u/\ell_p}}.
$$
Thus $\ell_p$ is the contour separation over which tangent alignment falls by the factor $e^{-1}$; it quantifies stiffness relative to thermal bending. The model bridges locally rod-like and globally coil-like conformations. It omits long-range self-avoidance in its simplest form, so sufficiently long dilute DNA can require excluded-volume corrections.
\b[Flexible and rigid limits.] Integrating the tangent correlation over the two contour positions gives the <end-to-end distance of a worm-like chain>:
$$
\langle R^2\rangle=2\int_0^L(L-u)e^{-u/\ell_p}du
=2\ell_pL-2\ell_p^2(1-e^{-L/\ell_p}).
$$
For a very flexible chain on the scale of its total length, $L/\ell_p\gg1$, the exponential is negligible and
$$
\boxed{\langle R^2\rangle\simeq2\ell_pL,\qquad R_{\mathrm{rms}}\simeq\sqrt{2\ell_pL}\ll L}.
$$
This is the <ideal chain> result with <Kuhn length> $b_K=2\ell_p$ and effective step count $N_K=L/(2\ell_p)$. Many persistent sections have decorrelated orientations. The flexible limit means a large ratio $L/\ell_p$ within the continuum model's range, not an extrapolation of the persistence length below all molecular length scales.
For a nearly rigid chain, $L/\ell_p\ll1$, the constant and linear terms in the exponential cancel those outside it. The <Taylor series> gives
$$
\langle R^2\rangle=L^2-\frac{L^3}{3\ell_p}+\frac{L^4}{12\ell_p^2}+\cdots,
$$
so
$$
\boxed{R_{\mathrm{rms}}\simeq L\left(1-\frac{L}{6\ell_p}+\cdots\right)\longrightarrow L}.
$$
This is the expected rod limit: all tangents are nearly aligned, and the end-to-end distance approaches the <arc length>.
\b[Helix–coil curves.] A <helix-coil transition model> allows ordered helical runs and disordered coil regions. Use $s$ and $\sigma$ with exactly the free-energy conventions defined in the paper. A negative $\Delta F_{\mathrm{hc}}$ gives $s>1$ and favors the helical state; a positive junction cost gives $0<\sigma<1$ and penalizes starting new domains. The supplied helical fraction is the long-chain form associated with the <Zimm-Bragg model>.
Its main features follow directly without assuming a sharp transition. Put $Q=(s-1)^2+4s\sigma$. For every $\sigma>0$,
$$
\boxed{\lim_{s\downarrow0}f_h(s)=0,\qquad f_h(1)=\frac12,\qquad\lim_{s\to\infty}f_h(s)=1},
$$
and differentiating gives
$$
\boxed{\frac{df_h}{ds}=\frac{\sigma(s+1)}{Q^{3/2}}>0,\qquad
\left.\frac{df_h}{ds}\right|_{s=1}=\frac1{4\sqrt\sigma}}.
$$
Thus all curves rise monotonically and cross at $(1,1/2)$. Increasing the boundary free-energy cost reduces $\sigma$, suppresses frequent coil–helix boundaries and makes longer runs favorable. The transition region becomes narrower and steeper: balancing $(s-1)^2$ against $4s\sigma$ gives a width of order $\sqrt\sigma$ near $s=1$.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2018-scm-helix-coil.png]
{title=Helical fraction and the sharpening of a helix-coil crossover}
{description=Original plots of the supplied fraction for several positive initiation parameters, with a magnified view near the common midpoint.}
For $\sigma=1$, the fraction simplifies to $\boxed{f_h=s/(1+s)}$, the noncooperative two-state result. For $0<\sigma\ll1$, the steep curve describes cooperative helix formation. In the singular limit $\sigma\downarrow0$, it approaches zero for $s<1$ and one for $s>1$; its midpoint remains $1/2$ when approached through positive $\sigma$. Setting $\sigma=0$ directly makes the formula indeterminate at $s=1$. For any fixed positive $\sigma$, the given long-chain curve is smooth: it is a cooperative crossover, not a true discontinuity. The plot varies $s$ at fixed $\sigma$; in an actual temperature scan both parameters may change, and a finite chain can further round the crossover.
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