#### Publication Date

5-2013

#### Advisor(s) - Committee Chair

Claus Ernst (Director), Uta Ziegler, Melanie Autin

#### Degree Program

Department of Mathematics

#### Degree Type

Master of Science

#### Abstract

In this thesis, we study the expected mean of the torsion angle of an n-step

equilateral random walk in 3D. We consider the random walk is generated within a confining sphere or without a confining sphere: given three consecutive vectors ^{→}*e*_{1} , ^{→}*e*_{2} , and ^{→}*e*_{3} of the random walk then the vectors ^{→}*e*_{1} and ^{→}*e*_{2} define a plane and the vectors ^{→}*e*_{2} and ^{→}*e*_{3} define a second plane. The angle between the two planes is called the torsion angle of the three vectors. Algorithms are described to generate random walks which are used in a particular space (both without and with confinement). The torsion angle is expressed as a function of six variables for a random walk in both cases: without confinement and with confinement, respectively. Then we find the probability density functions of these six variables of a random walk and demonstrate an explicit integral expression for the expected mean torsion value. Finally, we conclude that the expected torsion angle obtained by the integral agrees with the numerical average torsion obtained by a simulation of random walks with confinement.

#### Disciplines

Genetics and Genomics | Mathematics | Probability

#### Recommended Citation

He, Mu, "The Torsion Angle of Random Walks" (2013). *Masters Theses & Specialist Projects.* Paper 1242.

https://digitalcommons.wku.edu/theses/1242