2. Consider a system of identical but distinguishable particles, each of which has two states, with energica c and -c. Calculate: (i) the microcanonical partition function ?(E) of a system of N such particles with total fixed energy E; (ii) the canonical partition function of a system of / such particles, the total average energy (Eye, and the average energy per particle; (iii) the grand-canonical partition function, the average number of particles (N) = Noc, the total average energy (D)ec (express it in terms of Nog, () and the average energy per particle, that is (D)ac/Noc; (iv) Entropy: a) calculate the entropy in the microcanonical ensemble. Use Stirling approx- imation to simplify factorials and express Sm. in terms of x = E/(Nc) and N only. b) Find the entropy S, in the canonical ensemble and express it in terms of a = (E)./(Nc) and N only. Under which condition Sme ~ S,? c) Finally, calculate the entropy Sog in the grand-canonical ensemble and express it in terms of a = (F)cc/(Noot) and Neg only. In order to do so express Zoo in terms of Nec and then invert the expression for Nog to get Bu in terms of Noc, I. How does Sec compare to S,, 8, ? Under which condition they will be equivalent