# Internships (Stages)

Possibilities to write a master thesis, or to simply do bibliography research under my guidance exist at all times, both at the M1 and M2 level. I am happy to discuss possible projects centered around my research interestst or your interest with you. This is best discussed in person. Please contact me if you are interested.
Two current topics are listed below.

### Topic 1: Unzipping of RNA

One of the most exciting experimental developments at the frontier between physics and biology is our possiblity to attach the ends of an RNA or DNA molecule to beads, trap the beads in an optical tweezer, pull on the beads, and measure the ensuing forces as a function of bead-position (see figure). This force in the pico-Newton range contains a wealth of information on the biological system. It is also an exciting testing ground for theoretical concepts. In this internship, we propose to study the effects of thermal fluctuations onto the sawtooth like unzipping curves. Are these thermal effects negligible, or do they even change the large-scale physics?

### Topic 2: Extreme-Value Statistics for Fractional Brownian Motion

Brownian motion is a stochastic process which is Gaussian, scale invariant, translationally invariant, and Markovian, i.e. has independent increments. Apart from simple expectations, one is particularly interested in extremal properties: Does the process exceed a curtain threshold, and if yes, when? How long is the process prositive? This is important to correctly model the necessary capacity for dams, estimate the time one has to heat, etc.
While these questions can often be answered for Brownian motion, which is Markovian (a property essential in the solution),
many processes in nature are not Markovian, i.e. have memory.
Fractional Brownian motion (see figure) is such a generalization, indexed by the Hurst exponent *H*, and reducing for *H* = 1/2 to Brownian motion. For *H* < 1/2, the process is anticorrelated and rougher (red curves in the figure), while for *H* > 1/2 it is positively correlated.
We have recently been able to construct a systematic perturbative expansion around *H* = 1/2.
The stage proposes to study the limit of Hurst-index *H* = 1, and to try to build a systematic perturbation expansion in that limit.

Copyright (C) by Kay Wiese. Last edited October 25, 2016.