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• 1: Boltzmann entropy.
  A physical system in microcanonical equilibrium is equally likely to be in any of its microstates, and its entropy S is taken to simply be a function S = f(Ω) of the number of states Ω the system can be in. Show that in that case the additive character of S and the multiplicative character of Ω necessarily require that S be of the form k ln Ω, where k is a constant. Provide a detailed proof, showing all steps!

• 2: Ideal gas in a microcanonical state.
  Derive an expression for the entropy for a classical monatomic ideal gas of N particles of mass m, in a microcanonical state with total energy H(q, p) in the interval [E, E + Δ], with \(\Delta \ll E\). You may use the steps outlined in the lecture notes, but fill in and explain all of the steps not included there. Then show that the pressure equation of state for such a gas is pV = NkT.

• 3: Harmonic oscillators in a microcanonical state.
  Consider a system of two classical, uncoupled 1-dimensional simple harmonic oscillators, both of mass m and spring constant K, in a microcanonical state with energy H(q, p) in the interval [E, E + Δ], with \(\Delta \ll E\). Calculate the entropy of this system.
  

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