(b) Many connections have been made between entropy and classical information theory (which is actually the theory of information transfer). Shannon (1948) did some pioneering work in this regard by realizing that noisy information transfer (i.e. missing information) can be related to systems with higher entropy. He made the connection by realizing that for a system in contact with a heat bath at temperature T (Canonical Ensemble) the entropy can be written a: :PilnPi i=1 where B- is the probability that the system is in state 2'. Thus when many of the Pi's are small but nonzero (e.g., at high temperature) we have little information about the state of the system and the entropy is large. Shannon derived the above formula to measure the lost information in a message (a series of \"bits\") where B; was the probability that the \"bit\" is undisturbed. (i) (2 points) What is 13} in the Canonical Ensemble. (ii) (3 points) Derive the Shannon's expression for entropy above by assuming a system with nondegenerate accessible states EZ- : E1, E2, E3, . . . in contact with a heat bath at temperature T. [Hintc Our original denition of Helmholz Free Energy: F : U TO' might prove helpful]. (0) (3 points) Random Party Walk: Consider the movement of people in a crowded room of dancers as a random walk. Assume that the velocity of a partier is 1 m/s and that the mean free path between collisions is 0.5 m, with the nal direction randomized. How long does it take, on average, for a person starting in the middle of the room to reach one of the walls in a large circular room of diameter 40 111 compared to the time without collisions