Exercise #6

a) In the microcanonical ensemble consider the following Hamiltonian

$H = -h \sum_i \sigma_i$

where $\sigma_i=\pm 1$ and $i=1,N$.

  • Calculate the entropy $S(E,N,h)$ and plot it as a function of $E/Nh$.

  • Calculate the temperature defined as $1/T=\partial S/\partial E$ and explain why it can be negative.

b) Consider the previous Hamiltonian in the canonical ensemble.

  • Calculate the Helmlholtz free energy $F$ and determine its natural variables.

  • Evaluate the entropy $S$ and the internal energy $U$ and eliminate $T$ in favor of $U$ in the expression for $S$ to get $S(U,N,h)$. Compare with the previous expression and comment the result.

c) Consider the previous Hamiltonian in the grandcanonical ensemble.

  • Calculate the thermodynamic potential and determine its natural variables.

  • Calculate the average $<N>$ and its fluctuations and comment the results.