Thesis (PhD)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Several experiments [1, 2, 3, 4] have shown that e ective attractive interactions exist between con ned like-charged macromolecules. Theoretical approaches have not reached consensus as to precisely what the mechanism for the attraction is, but it is agreed that comprehending the role of the counterion arrangement around macromolecules is crucial for understanding the e ective macromolecule interactions. It is generally assumed that attraction only occurs in the limit of strong electrostatic coupling and is driven by correlation e ects that are neglible in a mean- eld approach, which is valid in the weak-coupling limit. However, in some experimental situations attraction occurs even in the limit of weak-coupling. We consider a eld-theoretic approach that includes uctuations to study the Coulomb interactions of con ned counterions with a single exible charged spherical macromolecule that can expand or collapse uniformly by changing its radius. We show how the linearised eld-theory (valid in the weak-coupling limit) is mapped onto the square-well potential of Quantum Mechanics. The con nement leads to bound states being present in the spectrum at all times. Bound states are non-perturbative and we investigate the role they play in the physics of the system. Some of the e ects are rather counter-intuitive. Firstly, upon expanding the macromolecule in a xed con nement volume, the uctuation part of the free energy favours a decrease in the free energy. Secondly, upon increasing the temperature to high but nite values, the uctuation contribution does not dominate the free energy as would be expected. The mathematical origins of these e ects are dicussed in detail and as part of the analysis we introduce a novel regularisation scheme for computing the functional determinant arising in the model considered where the cut-o is speci ed unambiguously in terms of physical parameters. / AFRIKAANSE OPSOMMING: Verskeie eksperimente [1, 2, 3, 4] toon dat makro-ione met gelyksoortige ladings, in `n eindige volume, `n e ektiewe aantrekkende krag ondervind. Alhoewel daar nog geen konsensus oor die presiese meganisme vir die aantrekking bereik is nie, is dit duidelik dat die rol van \counter-ion" rangskikking rondom die makro-ione belangrik is om die e ektiewe wisselwerkings te verstaan. Dit word algemeen aanvaar dat die aantrekkende krag slegs in die limiet van sterk elektrostatiese koppeling plaasvind en dat dit `n gevolg van \counter-ion" korrelasies is wat weglaatbaar is in `n gemiddelde veld benadering, wat geldig is in die limiet van swak elektrostatiese koppeling. Daar bestaan egter eksperimentele situasies waar die aantrekking in die limiet van swak elektrostatiese koppeling waargeneem word. Ons bestudeer die Coulomb wisselwerking tussen \counter-ions" en `n enkele rekbare sferiese makro-ioon vanuit `n veld-teoretiese beskouing wat uktuasies in ag neem. Die sferiese makro-ioon kan vergroot of verklein deur sy radius uniform te verander. Ons toon aan dat die gelineariseerde veldeteorie (geldig in die limiet van swak elektrostatiese koppeling) op die eindige-diepte put Kwantummeganiese model afgebeeld kan word. Die eindige volume van die sisteem het tot gevolg dat daar altyd gebonde toestande in die spektrum voorkom. Gebonde toestande is `n suiwer nie-steuringsteoretiese e ek en ons ondersoek die rol wat dit speel in die sika van die sisteem. Die teenwoordigheid van die gebonde toestande in die spektrum het `n paar teen-intuitiewe e ekte tot gevolg. Eerstens word die vrye energie verlaag soos die makro-ioon in `n eindige volume vergroot. Tweedens oorheers die uktuasie bydrae nie die vrye energie met toenemende temperatuur soos verwag sou word nie. Ons bespreek die wiskundige oorsprong van hierdie e ekte. As deel van die analise ontwikkel ons `n nuwe regulariseringstegniek vir die berekening van funksionaalintegrale waar die regulariseringsparameter ondubbelsinnig in terme van siese hoeveelhede uitgedruk kan word.

CAMARÃO, Diego de Lucena. Propriedades difusivas de sistemas clássicos confinados. 2011. 79 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2011. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-04-28T21:29:36Z No. of bitstreams: 1 2011_dis_dlcamarao.pdf: 2441556 bytes, checksum: ff4cb229929b646cece7ef667144d79d (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-04-29T17:54:40Z (GMT) No. of bitstreams: 1 2011_dis_dlcamarao.pdf: 2441556 bytes, checksum: ff4cb229929b646cece7ef667144d79d (MD5) / Made available in DSpace on 2015-04-29T17:54:41Z (GMT). No. of bitstreams: 1 2011_dis_dlcamarao.pdf: 2441556 bytes, checksum: ff4cb229929b646cece7ef667144d79d (MD5) Previous issue date: 2011 / Nesta dissertação, fizemos um estudo das propriedades difusivas de um sistema de partículas clássicas carregadas em canais quasi-unidimensionais. Mais especificamente, no Capítulo 2, apresentamos uma revisão do problema da difusão e do movimento browniano. Mostramos que as abordagens de Einstein e de Langevin para o movimento browniano são equivalentes no limite de tempos longos. Isto foi feito através do cálculo analítico do deslocamento quadrático médio (MSD) de um sistema unidimensional de N partículas não--interagentes através da solução da equação de difusão. No Capítulo 3, introduzimos o método de Dinâmica Molecular (DM), amplamente utilizado em simulações computacionais de sistemas de N partículas clássicas. Apresentamos dois métodos de integração numérica das equações de movimento: o algoritmo de Verlet e o algoritmo leapfrog. Abordamos brevemente o método de Dinâmica Molecular de Langevin (DML), que inclui um termo de flutuações térmicas (força estocástica), devido às colisões das moléculas do fluido com as partículas do sistema. Finalmente, apresentamos uma aproximação do método de DML chamada Dinâmica Browniana (DB). No Capítulo 4, estudamos as propriedades difusivas, através da análise do deslocamento quadrático médio, de um sistema de partículas clássicas carregadas sujeitas à ação de um potencial de confinamento unidimensional, analisando a transição do regime de difusão em linha (SFD) para o regime de difusão bidimensional (2D). Vimos como ocorre essa transição em função dos parâmetros que regulam o potencial de confinamento. Discutimos a validade dos resultados numéricos obtidos em relação a resultados analíticos teóricos encontrados na literatura. Finalmente, no Capítulo 5, apresentamos um resumo dos resultados obtidos, bem como discutimos perspectivas e sugestões para futuros trabalhos.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Nesta dissertaÃÃo, fizemos um estudo das propriedades difusivas de um sistema de partÃculas clÃssicas carregadas em canais quasi-unidimensionais. Mais especificamente, no CapÃtulo 2, apresentamos uma revisÃo do problema da difusÃo e do movimento browniano. Mostramos que as abordagens de Einstein e de Langevin para o movimento browniano sÃo equivalentes no limite de tempos longos. Isto foi feito atravÃs do cÃlculo analÃtico do deslocamento quadrÃtico mÃdio (MSD) de um sistema unidimensional de N partÃculas nÃo--interagentes atravÃs da soluÃÃo da equaÃÃo de difusÃo. No CapÃtulo 3, introduzimos o mÃtodo de DinÃmica Molecular (DM), amplamente utilizado em simulaÃÃes computacionais de sistemas de N partÃculas clÃssicas. Apresentamos dois mÃtodos de integraÃÃo numÃrica das equaÃÃes de movimento: o algoritmo de Verlet e o algoritmo leapfrog. Abordamos brevemente o mÃtodo de DinÃmica Molecular de Langevin (DML), que inclui um termo de flutuaÃÃes tÃrmicas (forÃa estocÃstica), devido Ãs colisÃes das molÃculas do fluido com as partÃculas do sistema. Finalmente, apresentamos uma aproximaÃÃo do mÃtodo de DML chamada DinÃmica Browniana (DB). No CapÃtulo 4, estudamos as propriedades difusivas, atravÃs da anÃlise do deslocamento quadrÃtico mÃdio, de um sistema de partÃculas clÃssicas carregadas sujeitas à aÃÃo de um potencial de confinamento unidimensional, analisando a transiÃÃo do regime de difusÃo em linha (SFD) para o regime de difusÃo bidimensional (2D). Vimos como ocorre essa transiÃÃo em funÃÃo dos parÃmetros que regulam o potencial de confinamento. Discutimos a validade dos resultados numÃricos obtidos em relaÃÃo a resultados analÃticos teÃricos encontrados na literatura. Finalmente, no CapÃtulo 5, apresentamos um resumo dos resultados obtidos, bem como discutimos perspectivas e sugestÃes para futuros trabalhos.

In this thesis, we have explored the commonalities and connections between different classes of quantum systems that do not thermalize. Specifically, we have (1) shown that localized systems possess conservation laws like integrable systems, which can be constructed in a systematic way and used to detect localization-delocalization transitions , (2) studied the phenomenon of many-body localization in a model with a single particle mobility edge, (3) shown that interesting finite-size scaling emerges, with universal exponents, when athermal quantum systems are forced to thermalize through the application of perturbations and (4) shown that these scaling laws also arise when a perturbation causes a crossover between quantum systems described by different random matrix ensembles. We conclude with a brief summary of each chapter. In Chapter 2, we have investigated the effects of finite size on the crossover between quantum integrable systems and non-integrable systems. Using exact diagonalization of finite-sized systems, we have studied this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L → ∞, non-integrability sets in for an arbitrarily small integrabilitybreaking perturbation. The crossover value of the perturbation scales as a power law ∼ L−3 when the integrable system is gapless and the scaling appears to be robust to microscopic details and the precise form of the perturbation. In Chapter 3, we have studied the crossover among different random matrix ensembles CHAPTER 6. CONCLUSION 127 [Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE)] realized in different microscopic models. We have found that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We have also found that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. Finally,we have conjectured that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system. In Chapter 4, we have outlined a procedure to construct conservation laws for Anderson localized systems. These conservation laws are found as power series in the hopping parameters. We have also obtained the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended depending on the strength of a coupling constant. We have formulated a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure for the Aubry-Andre model, we show that integrals of motion given by our construction are well-defined in the localized phase but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction. In Chapter 5, we have studied many body localization and investigated its nature in the presence of a single particle mobility edge. Employing the technique of exact diagonalization for finite-sized systems, we have calculated the level spacing distribution, time evolution of entanglement entropy, optical conductivity and return probability to characterize the nature of localization. The localization that develops in the presence of interactions in these systems appears to be different from regular Many-Body Localization (MBL) in that the growth of entanglement entropy with time is linear (like in CHAPTER 6. CONCLUSION 128 a thermal phase) instead of logarithmic but saturates to a value much smaller than the thermal value (like for MBL). All other diagnostics seem consistent with regular MBL