Problem set 1. (Total: 100 points) Consider an isolated system consisting of a large number N links of polymer chain located near a big cell membrane as shown in Fig. 1. Length of each link is d. For simplicity, let's consider only 1 dimensional motion on axis. Each link equally likely to be directed to the right and to the left. End of each link is electrically charged randomly so that electric dipole is induced on each link. (Note: the direction of electric dipole is conventionally defined from negative charge to positive charge.) The cell membrane is also electrically charged and produce a large electric field . Each link tends to align (but weakly due to temperature) to the electric field which results in the polymer folding to the left or to the right. The energy of each link can be expressed as = -p. . After folding the polymer chain, mean head-to-tail length is 1. Note) You need to show all procedures toward your answer to get a FULL credit. . Cell membrane Polymer Figure 1. Polymer near cell membrane. Both are electrically charged. (a) (30 points) What is the total number of states N2(l) with respect to mean head-to-tail length l? (b) (30 points) What is the total number of states N(E) with respect to the energy of the system? (c) (10 points) What is the entropy S as a function of the energy ? Please simplify the entropy by applying Stirling's formula In N! ~ N In N. (d) (10 points) What is the mean energy as a function of temperature T? (e) (20 points) What is the average force f at the folded polymer chain with length l as a function of temperature T? Please simplify your answer using large N. Hint) please use ensemble 2(1). In(1 + x) = x when x