STATISTICAL MECHANICS Exercises results 2025/2026

due date: October 21 2025

a) Consider the following thermodynamic cycle for a perfect gas:

b) Consider only one one-dimensional classical harmonic oscillator

now consider an ensemble made of replicas of the previous case i.e. one-dimensional classical harmonic oscillator with same $\omega$ and mass

c) 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.

d) 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.

due date: November 5th 2025

a) Read the notes on the virial theorem. Comment the action of repulsive or attractive interactions in the equation of state for imperfect gas.

b) Read the notes on fugacity expansion. 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 $z$ and eliminate the fugacity in favor of the average number of particle in order to have an approximation for the equation of state.

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

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

Compare and comments the two results.

c1) In a two state quantum system (say energy $E_0$ and $E_1$) a measure of energy can give with equal probability the two possible value of energy.

c2) An ensemble of quantum systems is made by a mixture of state $|0>$ (energy $E_0$) and $|1>$ (say energy $E_1$) with equal probability. Write the associated density matrix.

d) Evaluate the average energy as well as the average of the operator

$X=|0><1|+|1><0|$

in the two cases c1) and c2). Comments the results.

due date: November 18th 2025

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

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 < | x₁-x₂ | ² > as a function of x₀. Comment the results. Optional Redo the same calculations for Boltzmann’s particles.

b) Consider the perfect Fermi and Bose gas with a general single particle dispersion $\epsilon$(p)= a |p|^b Determine:

c) Consider a perfect gas of bosonic ultra-relativistic particles in 3 dimensions with vanishing chemical potential (black body). Calculate internal energy as a function of temperature and using the relation derived in the previous point prove that the radiation pressure is independent of volume and is proportional to T⁴.

d1) Read the notes Prove that the equilibrium single-particle density matrix for Boltzmann particle

is for a free particle

where Z⁽¹⁾ is the single particle partition function, \lambda is the De Broglie wavelength and |x>,|y> position eigenstates (in 3 dimensions). Find the normalization (A) for a tree dimensional space and the numerical coefficient (a). Comment the results. Compare the result for < x| $\rho$ |y > to that for <p₁| $\rho$ |p₂ > with p₁ and p₂ are three dimensional wave vectors.

d2) Optional Consider the equilibrium single-particle density matrix in the grand-canonical ensemble for massive non-relativistic non-interacting Bose particles. Write its form in momentum and position (3d) representation. Comment the limit \mu → 0^- for the density matrix in the coordinate representation. Using the fugacity expansion to all order or the integral expression of the density matrix write a code to evaluate numerically it when \mu < 0.