Exercise #1

due date: October 24 2022

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

ex_01.jpg

  • Draw the cycle in the T-S plane.

  • Calculate the total work exerted by the system (W).

  • Calculate the total heat exchanged by the system (Q).

  • Calculate the efficiency $\eta={W}/{Q_{abs}}$ where $Q_{abs}$ is the absorbed heat.

In the following you can alternatively choose b1) or b2).

b1) 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.

b2) 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.

  • When the system is chaotic the time average of $y$ converge to a time-independent value? Is this value independent on initial data?

c1) 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?

c2) 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

  • 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?