a) Consider the following Hamiltonian for $N$ independent spin $\sigma=\pm 1$

$H=-h\sum_i \sigma_i$

Perform the calculation in the microcanonical ensemble at fixed total energy, calculate the entropy and compare the results with that of canonical ensemble obtained in exercise #1 c).

b) Consider the problem on $N_1$ and $N_2$ non interprenetrating spheres disposed on a lattice of $M$ cells. Compute the dimensionless entropy of this system considering particles to be distinguishable in $1$ and $2$ flavors but indistinguishable in each given flavour.