Exercise #2

due date: 4th November 2019

a) deleted

b) Consider the BBGKY evolution equation for the reduced ensemble density of a classical system in the case of non intercating 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

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

Now remove the external potential and add a pairwise interaction between particles and assume that again velocities are thermalised such that the two particle distribution can be written as

$\rho^{(2)}(\vec{p}_1,\vec{p}_2\vec{q}_1,\vec{q}_2,t)=f(\vec{p}_1)f(\vec{p}_2)u^{(2)}(\vec{q}_1,\vec{q}_2,t)$

Derive the equation of motion for $u^{(2)}(\vec{q}_1,\vec{q}_2,t)$

c) Consider a system of classical non-interacting particles enclosed in a 2d circle of radius $R$. They are at equilibrium at temperature $T$. They are subject to a potential

$V(r)=\frac{1}{2}m\omega_0 {r^2}\;\;\;\;r<r_0$

$V(r)=\frac{1}{2}m\omega_0 {r^2}_0=const\;\;\;\;r>r_0$

and $r_o<R$

Determine and plot the density of the particle as a function of $r/\ell$ where $\ell$ is a suitably defined temperature dependent unit length.
Determine and plot the fraction of particle which are inside the smallest circle as a function of temperature.
For the two previous quantities discuss the limit $\ell <<r_0$ and $\ell >>r_0$