STATISTICAL MECHANICS Exercises results 2024/2025

due date: October 21 2024

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 2024

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 26th 2024

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.

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

You can choose between the following c1) and c2)

c1) 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.

c2) Following the same lines as done for classical interacting particles perform the fugacity expansion to the second order for quantum perfect gas of fermionic and bosonic particles and comment the results by comparison with classical interacting particles.

d1) 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.

due date: December 10th

a) In the mean field solution of the Ising model plot the isotherms $h(m)$ in the tree cases $T>T_c$,$T=T_c$,$T<T_c$. Plot the mean field estimate for the free energy $F(T,h)$ as a function of $h$ in the tree cases $T>T_c$,$T=T_c$,$T<T_c$ and discuss the results. I advice to use gnuplot which allows to define parametric functions.

b) Prove the isomorphism between the Ising model in the Canonincal Ensemble and the Lattice Gas model in the Grand-Canonical ensemble i.e. prove that $H_{LG}-\mu <N>$ maps onto $H_{Ising}$ up to some additive constants. After the solution of the previous point you will be able to plot the isotherms in the pressure-volume space for the lattice gas and discuss the results.

In the following you can alternatively choose c1) or c2).

c1) Read the notes about the Kac ring model:

c2) Read the notes about the Logistic map:

d) Read the chapter 3 of the textbook (Kerson Huang, "Statistical Mechanics", Second Edition, Wiley). Try to answer to one of the problems of the chapter at your convenience (or ask ChatGPT to do it…). If you dare you can try exercise 3.5 and for that it would be useful to read the notes on adiabatic expansion as well as this paper.

due date: January 7th 2025

a) Prove that in absence of interactions the Maxwell’s one particle density $\rho(p,q)=A \exp (-\beta (p^2/2m+\phi(q)))$ (where $\phi(q)$ is an external potential) is stationary solution of the BBGKY equation for $s=1$.

b) Following the lines of the notes derive the statistical properties of the noise term $\xi(t)$ as a result of the equilibrium distribution for the oscillators. Perform explicitly the $N\rightarrow \infty$ limit using the Ohmic spectral density.

c) Consider the stochastic equation for the moment of a particle under the action of external random forces $\xi(t)$ (in one dimension):

$\dot{p}(t)=-\gamma p(t) + \xi(t)$ + F

where

$\langle \xi(t) \rangle = 0$

$\langle \xi(t) \xi(t')\rangle = 2 M \gamma k_b T \delta(t-t')$

and $F$ is a constant in space and time external force.

d) The following point is optional

$\Delta(t)=\langle |x(t)-x(0)|^2 \rangle$