(a) Assuming that the total number of microstates accessible to a given statistical system is Ω, show that the entropy of the system, as given by eqn. (3.3.13), is maximum when all Ω states are equally likely to occur.
(b) If, on the other hand, we have an ensemble of systems sharing energy (with mean value Eˉ), then show that the entropy, as given by the same formal expression, is maximum when Pr∝exp(−βEr), β being a constant to be determined by the given value of Eˉ.
(c) Further, if we have an ensemble of systems sharing energy (with mean value Eˉ) and also sharing particles (with mean value Nˉ), then show that the entropy, given by a similar expression, is maximum when Pr,s∝exp(−αNr−βEs), α and β being constants to be determined by the given values of Nˉ and Eˉ.