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One-dimensional fluid model with oscillating, exponentially decaying pair interactions
Thesis (MSc)--Stellenbosch University, 2015.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2015
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| _version_ | 1867614038087696384 |
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| access_status_str | Open Access |
| author | Maziya, Gcina |
| author2 | Kastner, Michael |
| author_browse | Kastner, Michael Maziya, Gcina |
| author_facet | Kastner, Michael Maziya, Gcina |
| author_sort | Maziya, Gcina |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2015. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/97847 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:45:40.774Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2015 |
| publishDateRange | 2015 |
| publishDateSort | 2015 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/97847 One-dimensional fluid model with oscillating, exponentially decaying pair interactions Maziya, Gcina Kastner, Michael Stellenbosch University. Faculty of Science. Dept. of Physics. Phase transitions Statistical mechanics One-dimensional fluid Integral operators Transfer operators UCTD Thesis (MSc)--Stellenbosch University, 2015. ENGLISH ABSTRACT: For a one-dimensional fluid model where the pair interaction potential between the molecules consists of a hard core and an exponential attraction, Kac has shown that the partition function can be determined exactly in the thermodynamic limit (Kac, 1959). Kac concluded (after the necessary calculations) that there is no phase transition for such a system with the defined pair interaction potential. The aim of this study is to investigate the phase transition with a modified potential. Therefore we repeated part of the calculations of Kac up to the derivation of the so-called Kac integral equation (see, Appendix A). This enabled us to generalise the potential function of Kac to obtain a onedimensional fluid model with oscillating, exponentially decaying pair interactions. We then used the generalised potential to calculate the canonical partition function and obtained a modified version of the Kac integral equation. Unfortunately, we were unable to compute the eigenvalues of this modified integral equation with our chosen numerical method. Since knowing the eigenvalues is the only means to decide whether such a generalised model will show a phase transition, we were unable to investigate it. We conclude the eigenvalues have to be calculated using a different numerical method. AFRIKAANSE OPSOMMING: Kac (Kac, 1959) het gewys dat vir ’n een-dimensionele vloeistof model waar die paarinteraksie tussen die molekules bestaan uit ’n harde kern asook ’n eksponesieel aantrekkende deel bestaan, die partisiefunksie eksak in die termodinamiese limiet bereken kan word. Die gevolgtrekking van Kac se studie is dat daar nie ’n fase oorgang vir ’n sisteem met die tipe potensiaal bestaan nie. In hierdie studie ondersoek ons die fase oorgang met ’n veralgemeende potensiaal. Dus het ons die berekeninge in Kac se artikel tot en met die afleiding van die sogenaamde Kac integraal vergelyking herhaal (sien Byvoegsel A). Dit stel ons in staat om die Kac se potensiaalfunksie te veralgemeen na ’n model vir ’n een-dimensionele vloeistof met ’n ossilerende, eksponensieel dalende paar-interaksie tussen die molekules. Ons het toe die veralgemeende potensiaal gebruik om ’n gemodifiseerde weergawe van die Kac integraal vergelyking te bereken. Ongelukkig kon ons nie die eiewaardes van die gemodifiseerde integraal vergelyking met ons gekose numeriese metode bereken nie. Omdat die eiewaardes die enigste manier is om te bepaal of die fase oorgang sal plaasvind, al dan nie, kan ons ons nie daaroor uitspreek nie. Ons kon tot die gevolgtrekking dat die eiewaardes met ’n ander numeriese metode bereken sal moet word. 2015-12-14T07:42:46Z 2015-12-14T07:42:46Z 2015-12 Thesis http://hdl.handle.net/10019.1/97847 en_ZA Stellenbosch University viii, 46 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Phase transitions Statistical mechanics One-dimensional fluid Integral operators Transfer operators UCTD Maziya, Gcina One-dimensional fluid model with oscillating, exponentially decaying pair interactions |
| title | One-dimensional fluid model with oscillating, exponentially decaying pair interactions |
| title_full | One-dimensional fluid model with oscillating, exponentially decaying pair interactions |
| title_fullStr | One-dimensional fluid model with oscillating, exponentially decaying pair interactions |
| title_full_unstemmed | One-dimensional fluid model with oscillating, exponentially decaying pair interactions |
| title_short | One-dimensional fluid model with oscillating, exponentially decaying pair interactions |
| title_sort | one dimensional fluid model with oscillating exponentially decaying pair interactions |
| topic | Phase transitions Statistical mechanics One-dimensional fluid Integral operators Transfer operators UCTD |
| url | http://hdl.handle.net/10019.1/97847 |
| work_keys_str_mv | AT maziyagcina onedimensionalfluidmodelwithoscillatingexponentiallydecayingpairinteractions |