Exercise #1

due date: November 2nd 2020

You can alternatively choose a1) or a2) as a first point.

a1) Read the notes about the Kac ring model:

  • which set of variables describes a microscopic state?

  • which set of variables describes the macroscopic state?

  • write a code to calculate the number of black and white point (you can also do it considering a small size say N=4 and doing the evolution by hand…)

  • compare the output of the code with the "molecular-chaos" solution given in the notes and discuss the results.

a2) Read the notes about the Logistic map:

  • write a code to calculate the map evolution and the Lyapunov exponent.

  • discuss the results in the parameters space described in the note discussing the local stability and the Lyapunov exponent.

b1) Consider only one one-dimensional classical harmonic oscillator

  • write the Hamilton equation and plot a typical trajectory in the phase space

  • is the system chaotic?

b2) now consider an ensemble made of replicas of the previous case i.e. one-dimensional classical harmonic oscillator with same $\omega$ and mass

  • write the Liouville’s evolution for and ensemble of such systems using momenta and position as variables

  • write the Liouville’s evolution for and ensemble of such systems using action-angle variables (amplitude and phase)

  • consider an isoenergetic ensemble of oscillators (all osc. have energy=E). This ensemble starts with random phases between $\phi_0$ and $\phi_0+\Delta$. In such conditions evaluate $<x(t)>$. Does it tends to a constant value? Is the system ergodic? Is the system mixing?

c) Consider N classical independent one-dimensional harmonic oscillators (mass $m_i$, frequency $\omega_i$ $i=1,N$) whose initial data are taken according Boltzmann distribution for position and momenta at temperature $T$. Let them evolve and calculate the following correlation functions:

$C_{i,j}(t,t')=< p_i(t)p_j(t') >$, $G_{i,j}(t,t')=< x_i(t)x_j(t') >$ where averages and taken on the Boltzmann distribution of initial data.

By assuming a distribution of the oscillator’s frequencies ($\omega_i>0$) $P(\omega)\propto \omega^2 \;\;\; \omega<\Lambda$ as well as equal masses for all oscillators calculate in the large $N$ limit:

$C(t,t')=(1/N)\sum_{i,j} C_{i,j}(t,t')$, $G(t,t')=(1/N)\sum_{i,j} G_{i,j}(t,t')$

find the time evolution of these functions, temperature dependence and dependence on the cutoff $\Lambda$.

hint : express the solution for x and p as a function of initial data $x(0)$ and $p(0)$.

d) Consider quantum harmonic oscillators of frequency $\omega}$ in two distinct cases

  1. An ensemble of thermalized oscillators described by the density matrix $\hat{\rho}=\sum_n P_n |n><n|$ where $|n>$ are eigenstates of harmonic oscillators and $P_n\propto \exp(-\beta E_n)$

  2. A pure state $|\psi>=\sum_n \sqrt{P_n}|n>$ with the same $P_n$ as in i)

In both case calculate $<x^2>$ as a function of temperature - assuming a Boltzmann distribution for $P_n$ - and comment the results.