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	Author	Magnus, W.; Brosens, F.
	Title	Occupation numbers in a quantum canonical ensemble : a projection operator approach			Type	A1 Journal article
	Year	2019	Publication	Physica: A : theoretical and statistical physics	Abbreviated Journal	Physica A
	Volume	518	Issue	518	Pages	253-264
	Keywords	A1 Journal article; Theory of quantum systems and complex systems; Condensed Matter Theory (CMT)
	Abstract	Recently, we have used a projection operator to fix the number of particles in a second quantization approach in order to deal with the canonical ensemble. Having been applied earlier to handle various problems in nuclear physics that involve fixed particle numbers, the projector formalism was extended to grant access as well to quantum-statistical averages in condensed matter physics, such as particle densities and correlation functions. In this light, the occupation numbers of the subsequent single-particle energy eigenstates are key quantities to be examined. The goal of this paper is (1) to provide a sound extension of the projector formalism directly addressing the occupation numbers as well as the chemical potential, and (2) to demonstrate how the emerging problems related to numerical instability for fermions can be resolved to obtain the canonical statistical quantities for both fermions and bosons. (C) 2018 Elsevier B.V. All rights reserved.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication		Editor
	Language		Wos	000456359200021	Publication Date	2018-11-28
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0378-4371	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	2.243	Times cited	1	Open Access
	Notes	; ;			Approved	Most recent IF: 2.243
	Call Number	UA @ admin @ c:irua:157468			Serial	5223
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	Author	Magnus, W.; Lemmens, L.; Brosens, F.
	Title	Quantum canonical ensemble : a projection operator approach			Type	A1 Journal article
	Year	2017	Publication	Physica: A : theoretical and statistical physics	Abbreviated Journal	Physica A
	Volume	482	Issue		Pages	1-13
	Keywords	A1 Journal article; Theory of quantum systems and complex systems; Condensed Matter Theory (CMT)
	Abstract	Knowing the exact number of particles N, and taking this knowledge into account, the quantum canonical ensemble imposes a constraint on the occupation number operators. The constraint particularly hampers the systematic calculation of the partition function and any relevant thermodynamic expectation value for arbitrary but fixed N. On the other hand, fixing only the average number of particles, one may remove the above constraint and simply factorize the traces in Fock space into traces over single-particle states. As is well known, that would be the strategy of the grand-canonical ensemble which, however, comes with an additional Lagrange multiplier to impose the average number of particles. The appearance of this multiplier can be avoided by invoking a projection operator that enables a constraint-free computation of the partition function and its derived quantities in the canonical ensemble, at the price of an angular or contour integration. Introduced in the recent past to handle various issues related to particle-number projected statistics, the projection operator approach proves beneficial to a wide variety of problems in condensed matter physics for which the canonical ensemble offers a natural and appropriate environment. In this light, we present a systematic treatment of the canonical ensemble that embeds the projection operator into the formalism of second quantization while explicitly fixing N, the very number of particles rather than the average. Being applicable to both bosonic and fermionic systems in arbitrary dimensions, transparent integral representations are provided for the partition function Z(N) and the Helmholtz free energy F-N as well as for two- and four-point correlation functions. The chemical potential is not a Lagrange multiplier regulating the average particle number but can be extracted from FN+1 – F-N, as illustrated for a two-dimensional fermion gas. (C) 2017 Elsevier B.V. All rights reserved.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	Amsterdam	Editor
	Language		Wos	000405885500001	Publication Date	2017-04-20
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0378-4371	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	2.243	Times cited	1	Open Access
	Notes	; ;			Approved	Most recent IF: 2.243
	Call Number	UA @ lucian @ c:irua:145145			Serial	4722
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	Author	Sels, D.; Brosens, F.; Magnus, W.
	Title	Wigner distribution functions for complex dynamical systems : a path integral approach			Type	A1 Journal article
	Year	2013	Publication	Physica: A : theoretical and statistical physics	Abbreviated Journal	Physica A
	Volume	392	Issue	2	Pages	326-335
	Keywords	A1 Journal article; Theory of quantum systems and complex systems; Condensed Matter Theory (CMT)
	Abstract	Starting from Feynmans Lagrangian description of quantum mechanics, we propose a method to construct explicitly the propagator for the Wigner distribution function of a single system. For general quadratic Lagrangians, only the classical phase space trajectory is found to contribute to the propagator. Inspired by Feynmans and Vernons influence functional theory we extend the method to calculate the propagator for the reduced Wigner function of a system of interest coupled to an external system. Explicit expressions are obtained when the external system consists of a set of independent harmonic oscillators. As an example we calculate the propagator for the reduced Wigner function associated with the CaldeiraLegett model.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	Amsterdam	Editor
	Language		Wos	000311135200004	Publication Date	2012-09-14
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0378-4371;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	2.243	Times cited	9	Open Access
	Notes	; ;			Approved	Most recent IF: 2.243; 2013 IF: 1.722
	Call Number	UA @ lucian @ c:irua:101414			Serial	3921
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	Author	Sels, D.; Brosens, F.; Magnus, W.
	Title	On the path integral representation of the Wigner function and the BarkerMurray ansatz			Type	A1 Journal article
	Year	2012	Publication	Physics letters : A	Abbreviated Journal	Phys Lett A
	Volume	376	Issue	6/7	Pages	809-812
	Keywords	A1 Journal article; Theory of quantum systems and complex systems; Condensed Matter Theory (CMT)
	Abstract	The propagator of the Wigner function is constructed from the WignerLiouville equation as a phase space path integral over a new effective Lagrangian. In contrast to a paper by Barker and Murray (1983) [1], we show that the path integral can in general not be written as a linear superposition of classical phase space trajectories over a family of non-local forces. Instead, we adopt a saddle point expansion to show that the semiclassical Wigner function is a linear superposition of classical solutions for a different set of non-local time dependent forces. As shown by a simple example the specific form of the path integral makes the formulation ideal for Monte Carlo simulation.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	Amsterdam	Editor
	Language		Wos	000301167300005	Publication Date	2012-01-17
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0375-9601;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	1.772	Times cited	7	Open Access
	Notes	; ;			Approved	Most recent IF: 1.772; 2012 IF: 1.766
	Call Number	UA @ lucian @ c:irua:94006			Serial	2445
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	Author	Sels, D.; Brosens, F.; Magnus, W.
	Title	Classical trajectories : a powerful tool for solving tunneling problems			Type	A1 Journal article
	Year	2012	Publication	Physica: A : theoretical and statistical physics	Abbreviated Journal	Physica A
	Volume	391	Issue	1/2	Pages	78-81
	Keywords	A1 Journal article; Theory of quantum systems and complex systems; Condensed Matter Theory (CMT)
	Abstract	In the realm of Ehrenfests theorem, classical trajectories obeying Newtons laws have been proven useful to construct explicit solutions to the time-dependent WignerLiouville equation. Whereas previous works have particularly focused on the initial distribution function as a vehicle found to carry the signatures of quantum statistics into the time-dependent solution, the present paper shows that the LagrangeCharpit method based on classical trajectories can be successfully invoked as well to tackle quantum mechanical features with no classical counterpart, such as tunneling.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	Amsterdam	Editor
	Language		Wos	000297230700010	Publication Date	2011-08-25
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0378-4371;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	2.243	Times cited	7	Open Access
	Notes	; ;			Approved	Most recent IF: 2.243; 2012 IF: 1.676
	Call Number	UA @ lucian @ c:irua:92359			Serial	370
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	Author	Bertoni, G.; Verbeeck, J.; Brosens, F.
	Title	Fitting the momentum dependent loss function in EELS			Type	A1 Journal article
	Year	2011	Publication	Microscopy research and technique	Abbreviated Journal	Microsc Res Techniq
	Volume	74	Issue	3	Pages	212-218
	Keywords	A1 Journal article; Electron microscopy for materials research (EMAT); Theory of quantum systems and complex systems
	Abstract	Momentum dependent inelastic plasmon scattering can be measured by electron energy loss in a transmission electron microscope. From energy filtered diffraction, the characteristic angle of scattering and the cutoff angle are measured, using a thin film of aluminum as a model test. Rather than deconvolving the data (as done in previous works), a fitting technique is used to extract the loss function from angular resolved spectra, starting from a simple model simulation.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	New York, N.Y.	Editor
	Language		Wos	000288095200002	Publication Date	2010-07-06
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	1059-910X;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	1.147	Times cited	6	Open Access
	Notes	Fwo; Esteem; Iap; Goa			Approved	Most recent IF: 1.147; 2011 IF: 1.792
	Call Number	UA @ lucian @ c:irua:88782UA @ admin @ c:irua:88782			Serial	1222
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	Author	Brosens, F.; Magnus, W.
	Title	Newtonian trajectories : a powerful tool for solving quantum dynamics			Type	A1 Journal article
	Year	2010	Publication	Solid state communications	Abbreviated Journal	Solid State Commun
	Volume	150	Issue	43/44	Pages	2102-2105
	Keywords	A1 Journal article; Theory of quantum systems and complex systems; Condensed Matter Theory (CMT)
	Abstract	Since Ehrenfests theorem, the role and importance of classical paths in quantum dynamics have been examined by several means. Along this line, we show that the classical equations of motion provide a solution to quantum dynamics, if appropriately incorporated into the Wigner distribution function, exactly reformulated in a type of Boltzmann equation. Also the quantum-mechanical features of the canonical ensemble can be studied in this framework of Newtonian dynamics, if the initial distribution function is appropriately constructed from the statistical operator.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	New York, N.Y.	Editor
	Language		Wos	000284251700006	Publication Date	2010-09-20
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0038-1098;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	1.554	Times cited	7	Open Access
	Notes	; The authors thank J.T. Devreese and J. Tempere for interesting and helpful discussions, and, in particular, L.F. Lemmens for several valuable suggestions. One of the authors (F.B.) acknowledges the FWO projects G.0115.06 and G.0365.08 as well as the WOG project WO.033.09N, for financial support. ;			Approved	Most recent IF: 1.554; 2010 IF: 1.981
	Call Number	UA @ lucian @ c:irua:85795			Serial	2338
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	Author	Zhang, L.; Turner, S.; Brosens, F.; Verbeeck, J.
	Title	Model-based determination of dielectric function by STEM low-loss EELS			Type	A1 Journal article
	Year	2010	Publication	Physical review : B : condensed matter and materials physics	Abbreviated Journal	Phys Rev B
	Volume	81	Issue	3	Pages	035102
	Keywords	A1 Journal article; Electron microscopy for materials research (EMAT); Theory of quantum systems and complex systems
	Abstract	Dielectric properties of materials are crucial in describing the electromagnetic response of materials. As devices are becoming considerably smaller than the optical wavelength, the conventional measuring methods based on optical response are limited by their spatial resolution. Electron energy loss spectroscopy performed in a scanning transmission electron microscope is a good alternative to obtain the dielectric properties with excellent spatial resolution. Due to the overlap of diffraction discs in scanning transmission electron microscopy, it is difficult to apply conventional experimental settings to suppress retardation losses. In this contribution, a relativistic dielectric model for the loss function is presented which is used in a model based optimization scheme to estimate the complex dielectric function of a material. The method is applied to experiments on bulk diamond and SrTiO3 and shows a good agreement with optical reference data when retardation effects are included. Application of this technique to nanoparticles is possible but several theoretical assumptions made in the model of the loss function are violated and interpretation becomes problematic.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	Lancaster, Pa	Editor
	Language		Wos	000274002300027	Publication Date	2010-01-08
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	1098-0121;1550-235X;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	3.836	Times cited	9	Open Access
	Notes	Esteem – 026019; Fwo			Approved	Most recent IF: 3.836; 2010 IF: 3.774
	Call Number	UA @ lucian @ c:irua:81258UA @ admin @ c:irua:81258			Serial	2098
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	Author	Magnus, W.; Brosens, F.; Sorée, B.
	Title	Time dependent transport in 1D micro- and nanostructures: solving the Boltzmann and Wigner-Boltzmann equations			Type	A1 Journal article
	Year	2009	Publication	Journal of physics : conference series	Abbreviated Journal
	Volume	193	Issue	1	Pages	012004,1-012004,4
	Keywords	A1 Journal article; Condensed Matter Theory (CMT); Theory of quantum systems and complex systems
	Abstract	For many decades the Boltzmann distribution function has been used to calculate the non-equilibrium properties of mobile particles undergoing the combined action of various scattering mechanisms and externally applied force fields. When the latter give rise to the occurrence of inhomogeneous potential profiles across the region through which the particles are moving, the numerical solution of the Boltzmann equation becomes a highly complicated task. In this work we highlight a particular algorithm that can be used to solve the time dependent Boltzmann equation as well as its quantum mechanical extension, the WignerBoltzmann equation. As an illustration, we show the calculated distribution function describing electrons propagating under the action of both a uniform and a pronouncedly non-uniform electric field.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	Bristol	Editor
	Language		Wos	000277100400004	Publication Date	2009-11-17
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	1742-6596;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor		Times cited	2	Open Access
	Notes				Approved	Most recent IF: NA
	Call Number	UA @ lucian @ c:irua:82861			Serial	3667
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	Author	Magnus, W.; Brosens, F.; Sorée, B.
	Title	Modeling drive currents and leakage currents : a dynamic approach			Type	A1 Journal article
	Year	2009	Publication	Journal of computational electronics	Abbreviated Journal	J Comput Electron
	Volume	8	Issue	3/4	Pages	307-323
	Keywords	A1 Journal article; Condensed Matter Theory (CMT); Theory of quantum systems and complex systems
	Abstract	The dynamics of electrons and holes propagating through the nano-scaled channels of modern semiconductor devices can be seen as a widespread manifestation of non-equilibrium statistical physics and its ruling principles. In this respect both the devices that are pushing conventional CMOS technology towards the final frontiers of Moores law and the upcoming set of alternative, novel nanostructures grounded on entirely new concepts and working principles, provide an almost unlimited playground for assessing physical models and numerical techniques emerging from classical and quantum mechanical non-equilibrium theory. In this paper we revisit the Boltzmann as well as the WignerBoltzmann equation which offers a valuable platform to study transport of charge carriers taking part in drive currents. We focus on a numerical procedure that regained attention recently as an alternative tool to solve the time-dependent Boltzmann equation for inhomogeneous systems, such as the channel regions of field-effect transistors, and we discuss its extension to the WignerBoltzmann equation. Furthermore, we pay attention to the calculation of tunneling leakage currents. The latter typically occurs in nano-scaled transistors when part of the carrier distribution sustaining the drive current is found to tunnel into the gate due the presence of an ultra-thin insulating barrier separating the gate from the channel region. In particular, we discuss the paradox related to the very existence of leakage currents established by electrons occupying quasi-bound states, while the (real) wave functions of the latter cannot carry net currents. Finally, we describe a simple model to resolve the paradox as well as to estimate gate currents provided the local carrier generation rates largely exceed the tunneling rates.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	S.l.	Editor
	Language		Wos	000208236100009	Publication Date	2009-09-28
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	1569-8025;1572-8137;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	1.526	Times cited	4	Open Access
	Notes				Approved	Most recent IF: 1.526; 2009 IF: NA
	Call Number	UA @ lucian @ c:irua:89503			Serial	2110
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	Author	Brosens, F.; Magnus, W.
	Title	Carrier transport in nanodevices: revisiting the Boltzmann and Wigner distribution functions			Type	A1 Journal article
	Year	2009	Publication	Physica status solidi: B: basic research	Abbreviated Journal	Phys Status Solidi B
	Volume	246	Issue	7	Pages	1656-1661
	Keywords	A1 Journal article; Condensed Matter Theory (CMT); Theory of quantum systems and complex systems
	Abstract	In principle, transport of charged carriers in nanometer sized solid-state devices can be fully characterized once the non-equilibrium distribution function describing the carrier ensemble is known. In this light, we have revisited the Boltzmann and the Wigner distribution functions and the framework in which they emerge from the classical respectively quantum mechanical Liouville equation. We have assessed the method of the characteristic curves as a potential workhorse to solve the time dependent Boltzmann equation for carriers propagating through spatially non-uniform systems, such as nanodevices. In order to validate the proposed solution strategy, we numerically solve the Boltzmann equation for a one-dimensional conductor mimicking the basic features of a biased low-dimensional transistor operating in the on-state. Finally, we propose a computational scheme capable of extending the benefits of the above mentioned solution strategy when it comes to solve the Wigner-Liouville equation.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	Berlin	Editor
	Language		Wos	000268659100033	Publication Date	2009-04-22
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0370-1972;1521-3951;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	1.674	Times cited	8	Open Access
	Notes				Approved	Most recent IF: 1.674; 2009 IF: 1.150
	Call Number	UA @ lucian @ c:irua:77953			Serial	284
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	Author	Shanenko, A.A.; Tempère, J.; Brosens, F.; Devreese, J.T.
	Title	Mesoscopic samples: the superconducting condensate via the Gross.Pitaevskii scenario			Type	A1 Journal article
	Year	2004	Publication	Solid state communications	Abbreviated Journal	Solid State Commun
	Volume	131	Issue		Pages	409-414
	Keywords	A1 Journal article; Theory of quantum systems and complex systems; Condensed Matter Theory (CMT)
	Abstract
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	New York, N.Y.	Editor
	Language		Wos	000223011700012	Publication Date	2004-03-20
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0038-1098;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	1.554	Times cited	1	Open Access
	Notes				Approved	Most recent IF: 1.554; 2004 IF: 1.523
	Call Number	UA @ lucian @ c:irua:48282			Serial	2000
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	Author	Brosens, F.; Fomin, V.M.; Lemmens, L.; Peeters, F.M.
	Title	Liber amicorum in honour of Jozef T. Devreese			Type	ME3 Book as editor
	Year	2003	Publication		Abbreviated Journal
	Volume		Issue		Pages
	Keywords	ME3 Book as editor; Theory of quantum systems and complex systems; Condensed Matter Theory (CMT)
	Abstract
	Address
	Corporate Author				Thesis
	Publisher	Wiley	Place of Publication	Weinheim	Editor
	Language		Wos		Publication Date	0000-00-00
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN		ISBN		Additional Links	UA library record; WoS full record;
	Impact Factor		Times cited		Open Access
	Notes				Approved	Most recent IF: NA
	Call Number	UA @ lucian @ c:irua:43416			Serial	1817
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	Author	Backes, W.H.; Peeters, F.M.; Brosens, F.; Devreese, J.T.
	Title	Dispersion of longitudinal plasmons for a quasi-two-dimensional electron gas			Type	A1 Journal article
	Year	1992	Publication	Physical review : B : condensed matter and materials physics	Abbreviated Journal	Phys Rev B
	Volume	45	Issue	15	Pages	8437-8442
	Keywords	A1 Journal article; Condensed Matter Theory (CMT); Theory of quantum systems and complex systems
	Abstract	Confinement of electrons in ultrathin metallic films leads to subbands. By increasing the thickness of the electron layer, the subbands will dissolve into a quasicontinuum, with the number of electrons per unit volume kept constant. Within the random-phase approximation, the two-dimensional plasmon, which originally follows Stern's dispersion relation, becomes a longitudinal surface plasmon. The plasmon excitations of a model metallic film are investigated by including all subbands. Single-particle excitations, which exhibit the depolarization shift, converge into the plasma excitation spectrum. With further increases in the film thickness, the bulk plasmon arises and the surface plasmon remains. Our analysis shows how quantum size effects evolve into hydrodynamical classical size effects with increasing thickness of the film.
	Address
	Corporate Author				Thesis
	Publisher		Place of Publication	Lancaster, Pa	Editor
	Language		Wos	A1992HR33600028	Publication Date	2002-07-27
	Series Editor		Series Title		Abbreviated Series Title
	Series Volume		Series Issue		Edition
	ISSN	0163-1829;1095-3795;	ISBN		Additional Links	UA library record; WoS full record; WoS citing articles
	Impact Factor	3.736	Times cited	37	Open Access
	Notes				Approved	no
	Call Number	UA @ lucian @ c:irua:2738			Serial	737
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