Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT:In this thesis we discuss some of the anomalies present in systems with long-range interactions,
for instance negative speci c heat and negative magnetic susceptibility, and show how
they can be related to the convexity properties of the thermodynamic potentials and nonequivalence
of ensembles. We also discuss the possibility of engineering long-range quantum
spin systems with cold atoms in optical lattices to experimentally verify the existence of nonequivalence
of ensembles. We then formulate an expression for the density of states when
the energy and magnetisation correspond to a pair of non-commuting operators. Finally we
analytically compute the entropy s( ;m) as a function of energy, , and magnetisation, m, for
the anisotropic Heisenberg model with Curie-Weiss type interactions. The results show that
the entropy is non-concave in terms of magnetisation under certain circumstances which in
turn indicates that the microcanonical and canonical ensembles are not equivalent and that
the magnetic susceptibility is negative. After making an appropriate change of variables we
show that a second-order phase transition can be present at negative temperatures in the
microcanonical ensemble which cannot be represented in the canonical ensemble. / AFRIKAANSE OPSOMMING: In hierdie tesis bespreek ons van die onverwagte eienskappe wat sisteme met lang afstand wisselwerkings
kan openbaar, byvoorbeeld negatiewe spesi eke warmte en negatiewe magnetiese
suseptibiliteit. Ons dui ook die ooreenkoms tussen hierdie gedrag en die konveksiteit van
die termodinamiese potensiale en nie-ekwivalente ensembles aan. Hierna bespreek ons die
moontlikheid om lang afstand kwantum spin sisteme te realiseer met koue atome in 'n optiese
rooster. Daarna wys ons hoe dit moontlik is om 'n uitdrukking vir die digtheid van toestande
te formuleer vir sisteme waar die energie en magnetisasie ooreenstem met operatore wat nie
met mekaar kommuteer nie. Uiteindelik bepaal ons die entropie, s( ;m), in terme van die
energie, , en magnetisasie, m, vir die anisotropiese Heisenberg model met Curie-Weiss tipe
interaksies. Die resultate wys dat die entropie onder sekere omstandighede nie konkaaf in
terme van magnetisasie is nie. Dit, op sy beurt, dui aan dat die mikrokanoniese en kanoniese
ensembles nie ekwivalent is nie en dat die magnetiese suseptibiliteit negatief kan wees.
Nadat ons 'n toepaslike transformasie van veranderlikes maak, wys ons dat 'n tweede orde
fase-oorgang by negatiewe temperature kan plaasvind in die mikrokanoniese ensemble wat nie
verteenwoordig kan word in die kanoniese ensemble nie.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/20003 |
| Date | 03 1900 |
| Creators | Olivier, G. J. F. (Gerrit Jacobus Francois) |
| Contributors | Kastner, Michael, Stellenbosch University. Faculty of Science. Dept. of Physics. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 61 p. : ill. |
| Rights | Stellenbosch University |
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