2. Consider a system of identical but distinguishable particles, each of which has two states, with energies and - . Calculate (i) the microcanonicalpartitionfunction ?( E) of a system of A such particles with total fixed energy E; (ii) the canonical partitionfunctionof a systemof A such particles, the total average energy (E) , and the average energy per particle (iii) the grand canonical partitionfunction the average number of particles (N) = Noc the total average energy (E) (express it in termsof Nep ) and the average energy per particle thatis (F)cc cc: (iv) Entropy: a) calculate the entropy in the microcanonical ensemble. Use Stirling approx- imationto simplify factorialsand express S, in termsof a = E/ (N ) and N only. b) Find the entropy S, in the canonical ensemble and express it in terms of a = (E)/(N ) and N only. Under which condition S, ' S ? c) Finally, calculate the entropy Soc in the grand canonical ensemble and express it in termsof a = (B)go/(Ng ) and Na, only. In order to do so express Z in termsof Nap and then invert the expression for Nae to get 8 p in terms of Nap, I. How does So compareto S., S ? Under which condition they will be equivalent