A simple system in which to study entropy and the Boltzmann distribution consists of N independent two-state subsystems coupled thermally to a reservoir at temperature T. The two states in each subsystem have energies 0 (by definition) and ϵ. Find the partition function, the internal energy, and the entropy of the total system as a function of temperature. Explain why the internal energy approaches Nϵ/2 as T → ∞. Is this what you expected? It may help if you explain why the entropy approaches Nk_B ln 2 in the same limit.