|Due: March 14, Extended: March 28:
(1) Read Nelson Chapter 9
(2) Compare the force extension curves of simulated Worm Like Chains (WLCs) to the approximate functional form given by Nelson eq. 9.42. Assume a link size of 1 nm, persistence length of 50 nm and that the applied force is along the z axis.
(a) Given a link with direction r(θ,φ), where θ is polar angle from the z-axis, find the probability of the next link r'(θ,φ,θ',φ').
(b) Show how to make a random selection of θ',φ' from the probability distribution in (a) when given random numbers uniformly distributed from 0 to 1.
(c) Use (a) and (b) to construct a 1000 link chain for a certain force.
(d) Repeat (c) as many times as necessary for the same force to obtain a well defined <z>.
(e) Repeat (d) for several different forces between 0 and the force that gives <z>~Lo where Lo is the T=0 K length.
(f) Plot <z> vs f of your simulation with errors given by standard error of the mean. On the same plot also show eq. 9.42. Be sure to include correct units.
(g) Comment on your result.