due date: 19th November 2019

a) Consider the stochastic equation for the moment of a particle under the action of external random forces $\xi(t)$ (in one dimension):

$\dot{p}(t)=-\gamma p(t) + \xi(t)$

where

$\langle \xi(t) \rangle = 0$

$\langle \xi(t) \xi(t')\rangle = 2 M \gamma k_b T \delta(t-t')$

$\Delta(t)=\langle |x(t)-x(0)|^2 \rangle$

b) A polymer can be constructed as a three dimensional random walk where the position of the n+1-th monomer is given by

$\vec{r}_{n+1}=\vec{r}_{n}+a\hat{u}_{n+1}$

where $a$ is the monomer spacing and $\hat{u}$ is a random unit vector. The length of the polymer made by $N+1$ monomers can be estimated by

$\ell= \sqrt{\langle |\vec{r}_{N}-\vec{r}_{0}|^2 \rangle}}$

where the average is taken over the random orientation of the unit vectors $\hat{u}_n$.

c) Prove that the entropy $S$ is given by $S = -k_B tr \rho \log \rho$ where $\rho$ is the equilibrium density matrix in the canonical ensemble. First perform the calculation in the classical canonical ensemble, then generalize it to the quantum case.

d) Consider the Fokker-Planck equation in one dimension

$\frac{\partial}{\partial t} P(x,t) = \frac{\partial}{\partial x} F(x)P(x,t) +\frac {\epsilon}{2}\frac{\partial^2}{\partial x^2} P(x,t)$

following the lines of the notes

show that for a potential problem

$F(x) = -\frac{\partial}{\partial x} V(x)$

the stationary distribution is given by

$P(x)=\exp(-2V(x)/\epsilon) $.

Derive the Maxwell Boltzmann distribution as stationary distribution of the momenta and of the position. For the position distribution use the overdamped approximation in the corresponding Langevin equation.