{"id":{"repo_id":"wku-diss","oai_identifier":"oai:digitalcommons.wku.edu:theses-2245"},"canonical_url":"https://search.dev.ndltd.org/etd/wku-diss/oai:digitalcommons.wku.edu:theses-2245","repository":{"repo_id":"wku-diss","name":"Western Kentucky University","base_url":"https://digitalcommons.wku.edu/do/oai/"},"display":{"title":"The Torsion Angle of Random Walks","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 →e1 , →e2 , and →e3 of the random walk then the vectors →e1 and →e2 define a plane and the vectors →e2 and →e3 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.

","abstract_html":"<p>In this thesis, we study the expected mean of the torsion angle of an n-step<br />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 <sup>→</sup><em>e</em><sub>1</sub> , <sup>→</sup><em>e</em><sub>2</sub> , and <sup>→</sup><em>e</em><sub>3</sub> of the random walk then the vectors <sup>→</sup><em>e</em><sub>1</sub> and <sup>→</sup><em>e</em><sub>2</sub> define a plane and the vectors <sup>→</sup><em>e</em><sub>2</sub> and <sup>→</sup><em>e</em><sub>3</sub> 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.</p>","abstract_has_math":false,"creators":["He, Mu"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics","degree_department":null,"school":null,"contributors":["Claus Ernst (Director), Uta Ziegler, Melanie Autin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-01T07:00:00Z","date_published":"2013-05-01T07:00:00Z","updated_at":"2026-07-24T06:08:16Z","subjects":["Coordinates","Integration","Probabilities","Polymers","Polygons","Torsion","DNA Viruses","Genetics and Genomics","Mathematics","Probability"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/1242","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Claus Ernst (Director), Uta Ziegler, Melanie Autin"]},{"key":"dc:creator","label":"Author","values":["He, Mu"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Coordinates","Integration","Probabilities","Polymers","Polygons","Torsion","DNA Viruses","Genetics and Genomics","Mathematics","Probability"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/1242"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["

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 →e1 , →e2 , and →e3 of the random walk then the vectors →e1 and →e2 define a plane and the vectors →e2 and →e3 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.

"]},{"key":"dc:title","label":"Title","values":["The Torsion Angle of Random Walks"]}]}],"canonical_facts":{"dc:contributor":["Claus Ernst (Director), Uta Ziegler, Melanie Autin"],"dc:creator":["He, Mu"],"dc:description.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 →e1 , →e2 , and →e3 of the random walk then the vectors →e1 and →e2 define a plane and the vectors →e2 and →e3 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.

"],"dc:identifier":["https://digitalcommons.wku.edu/theses/1242"],"dc:subject":["Coordinates","Integration","Probabilities","Polymers","Polygons","Torsion","DNA Viruses","Genetics and Genomics","Mathematics","Probability"],"dc:title":["The Torsion Angle of Random Walks"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:08:16Z"}