Exercise #3

a) $N$ particles of a perfect gas are thermalized at temperature $T$. A force $F$ is exerted on a piston of mass $M$ along the $x$ axis as in the figure.

piston.jpg

  • Write the Hamiltonian of the gas and the piston.

  • Calculate the distribution of the position $x$ of the piston at equilibrium, its average value and its fluctuation.

  • Calculate the thermodynamic potential and its first and second derivative w.r.t. $F$.

  • Discuss the results obtained in the last two points.

b) Consider $N$ independent harmonic oscillators of equal mass $m$ and different frequencies $\omega_i$ ($i=1,N$).

  • Consider one general initial data and calculate the time everage of the kinetic energy and proof that it is equals to half of the total energy ($E$) for any initial data.

  • Consider two different ensembles of initial data being $\epsilon_i$ the initial total energy of a given oscillator

    • $\epsilon_1=E$, $\epsilon_i=0$ if ($i=2,N$)

    • $\epsilon_i=E/N$ ($i=1,N$)

proof that the ensemble average of the kinetic energy equals the time average in both cases.