The relation between force and extension for a random walk “macromolecule” in three
(a) In class, we have derived the relation between force and extension for a one-dimensional random walk “macromolecule”. Derive the analogous relation between force and extension for a three-dimensional random walk “macromolecule”, in which each monomer is permitted to point in one of six directions. Make a plot of the resulting function. Hint: find the average length of one segment in the direction of the force, <L0>, and then the average length for the entire molecule is N times as large, N<L0>.
(b) In the small-force limit, the force-extension curve is linear; that is, in this regime, the polymer behaves like an ideal Hookean spring with a stiffness constant k ∝kBT/Ltota
Demonstrate this claim and deduce the numerical factor that replaces the proportionality with a strict equality.