dc.rights.license | CC-BY-NC-ND | |
dc.contributor.advisor | Morais Smith, C. | |
dc.contributor.advisor | Zegeling, P.A. | |
dc.contributor.advisor | Ozela, R.F. | |
dc.contributor.author | Verstraten, R.C. | |
dc.date.accessioned | 2020-07-27T18:00:30Z | |
dc.date.available | 2020-07-27T18:00:30Z | |
dc.date.issued | 2020 | |
dc.identifier.uri | https://studenttheses.uu.nl/handle/20.500.12932/36326 | |
dc.description.abstract | We begin with an overview of fractional derivatives, which have many different definitions, not all of which are equivalent. For some of the most commonly used definitions, we present a few properties and techniques for solving fractional differential equations. Furthermore, we show some of the key differences when solving identical equations using a different definition. There are already applications of fractional derivatives, but each application requires a critical assessment for which definition is most suitable. We show a new application of fractional derivatives in the field of glasses, making use of Caputo fractional derivatives. An analytical solution of the fractional Langevin equation is obtained, where the first-order friction term is replaced by a Caputo fractional derivative of order s. Then, we show that for 0<s<0.1, the ground state of the fractional Langevin solutions exhibits emergent periodic glassy behaviour, thus characterising the recently conjectured time glass. Finally, we present a semi-classical microscopic model, which, in the low-temperature limit, is effectively described by the fractional Langevin equation, thus establishing the link between sub-ohmic open systems and fractional derivative equations. | |
dc.description.sponsorship | Utrecht University | |
dc.format.extent | 4766986 | |
dc.format.mimetype | application/pdf | |
dc.language.iso | en | |
dc.title | The fractional Langevin equation | |
dc.type.content | Honours Program Thesis | |
dc.rights.accessrights | Open Access | |
dc.subject.keywords | Fractional, Calculus, Caputo, Derivative, Time, Glass, Langevin, sub-ohmic, open systems, Applications, Overview, Brownian motion, Emergent periodicity | |
dc.subject.courseuu | Theoretical Physics | |