Exercise #2

due date: November 15 2021

a) 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 $p(0)$.

b) 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.

c) Consider the BBGKY evolution equation for the reduced ensemble density of a classical system in the case of non interacting particles but in the presence of an external potential $\Phi(\vec{q})$.

Make the hypotesis that velocities are thermalised such that the one particle distribution can be written as

$\rho^{(1)}(\vec{p},\vec{q},t)=f(\vec{p})u^{(1)}(\vec{q},t)$

Demonstrate that the equilibrium distribution for the positional part is

$u^{(1)}(\vec{q})\propto \exp [-\beta \Phi(\vec{q}) $ ]

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. If you dare you can try exercise 3.5 and for that it would be useful to read this paper