Problem 2 from PSET 3:

Problem 2: statistical mechanics of two-level systems in the canonical ensemble. We will now address problem 3 of pset 2 using the canonical ensemble. Consider N distin- guishable particles, each of which can be in a state with energy 0 or E. (a) Defining E = ke, with k = 0,.... N, express the partition function Z(T) as a sum over all possible k. Use the binomial expansion to simplify the partition function. (b) Compute the average (E)y and plot it as a function of T. Analyze the limits T -+ 0 and T -+ co. (c) Invert the relation found in part (b) and express T as a function of E. Does it match with the expression you found in pset 2? (d) Compute the energy variance of = (E?)T -(E)?, and then find og/(E). How does the latter scale with N? (e) Take N = 10, and plot I(Eje-E/ as a function of r = E/(NE) for T = 0.256. Repeat the plot for N = 20. How does the mean and variance of the distribution change?Problem 3: Systems with negative temperature. A system contains / distinguishable particles, each of which can be either in a state with energy 0 or E. a. Compute the number of configurations I'(E) that have total energy E = N.e, with N. <.n. b. plot the entropy s="logI(E)]" as a function of e. determine sign temperature when e> NE/2. c. How do you interpret the results of point b? If a system with negative temperature (E > Ne/2) is put in contact with another system with positive temperature (E