due date: December 14th 2020

a) Consider a classical perfect gas and calculate the entropy in the canonical ensemble. Compare the result with that given, in the microcanonical ensemble, by the Sackur-Tetrode formula and show the ensemble equivalence in the thermodynamic limit.

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

$H=-gB\sum_i \sigma_i$

Where $B$ is the external magnetic field along $z$, $g$ a coupling constant and $\sigma_i$ the Pauli matrix $\sigma_z$ at a given site $i$. Spin operators at different sites commute.

Perform the calculation in the microcanonical ensemble at fixed total energy, calculate the entropy. Plot the dimensionless entropy per spin ($S/k_B N$) as a function of a suitably defined dimensionless energy.

Perform the calculation of the entropy in the canonical ensemble at fixed total temperature.

Compare the results of the previous two points by expressing the canonical entropy as a function of the internal energy.

Comment the result.

c) In the grandcanonical ensemble calculate the dimensionless ratio $\beta P/\rho$ where $\rho=N/V$ is the number density by performing a fugacity expansion in $z=\exp(\beta\mu)$ up to second order in $z$.

Perform the calculation in the case of an interacting classical gas with pair interactions.

Perform the calculation in the case of a quantum perfect gas in the case of Fermi-Dirac and Bose-Einstein statistics.

Express the result in term of the number density in both cases. Comment the two results.

d) Consider two one particle states made by one-dimensional gaussians

$g_1(x)=A\left ( \exp(-\frac{(x-x_0)}{2\sigma^2}) \right)^{1/2}$

$g_2(x)=A\left ( \exp(-\frac{(x+x_0)}{2\sigma^2}) \right)^{1/2}$

with $A$ being a normalisation constant. Using these states write a possible state for 2 fermions and 2 bosons in state $1$ or $2$ neglecting the spin component. Calculate the average distance $<(x_1-x_2)^2>$ as a function of $x_0$ (you can choose $\sigma^2=1$). Comment the results.