(a) Show that, for two large systems in thermal contact, the number \(\Omega^{(0)}\left(E^{(0)}, E_{1}ight)\) of Section 1.2 can be expressed as a Gaussian in the variable \(E_{1}\). Determine the root-mean-square deviation of \(E_{1}\) from the mean value \(\bar{E}_{1}\) in terms of other quantities pertaining to the problem.

(b) Make an explicit evaluation of the root-mean-square deviation of \(E_{1}\) in the special case when the systems \(A_{1}\) and \(A_{2}\) are ideal classical gases.