4. (30) Consider a system with \( N=3300 \) particles and \( U=4510 \). There are 3 energy levels \( \varepsilon_{1}=1, \varepsilon_{2}=2 \), and \( \varepsilon_{3}=3 \). The degeneracy is the same for all levels, and \( g_{i} \gg N \).
a) Determine the most probable macrostate \( \mathrm{N}_{1, \text { mp }}, \mathrm{N}_{2} \), mp, \( \mathrm{N}_{3, \text { mp }} \).
b) Show that macrostates with \( \mathrm{N}_{1}=\mathrm{N}_{1, \mathrm{mp}}+1 \) and \( \mathrm{N}_{1}=\mathrm{N}_{1, \mathrm{mp}}-1 \) are less probable than the most probable macrostate you rounded to in part (a) by showing \( \ln (\mathrm{W}) \) for those macrostates is less than \( \ln \left(W_{m p}\right) \). Recall from Boltzmann that \( \ln (W)=\ln \left[\left(\Pi\left(g_{i}{ }^{N} / N i !\right)\right]\right. \). In this case, \( \Pi_{\mathrm{i}}{ }^{N i}=g^{N} \) since \( g_{i} \) are all same.
