Progress in Nonequilibrium Green's Functions II

Preface.

CONTENTS.

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No abstract received.

A brief review of early Russian works on the Green’s functions applications to many body theory, particularly for nonequilibrium states and processes, is presented. Discussed are some general features and relations of the real-time Nonequilibrium Green’s function (NGF) matrices method to some other approaches.

A model of superconductivity in high-temperature superconducting layered cuprates is proposed, based on the extended saddle point singularities in the electron spectrum, weak screening of the Coulomb interaction and phonon-mediated interaction between electrons plus a small short-range repulsion of Hund’s, or spin-fluctuation, origin. This permits to explain the large values of T_c, features of the isotope effect on oxygen and copper, the existence of two types of the order parameter, the peak in the inelastic neutron scattering, the positive curvature of the upper critical field, as function of temperature, etc.

The resonant tunneling mechanism for the c-axis transport is proposed. Real physical properties are calculated and compared with experimental data. These include the temperature dependence of the static c-axis conductivity in the normal state, frequency dependence of the optical conductivity and stationary supercurrent along the c-axis. It is demonstrated that for the latter the coherence of resonant tunneling through different centers is of primary importance.

The resonant tunneling idea is used for description of the origin and some properties of the “pseudogap phase”. The superconducting critical temperature in this picture is defined at low doping by establishment of a 3-dimensional phase correlation between the layers, and at high doping by destruction of a d-wave superconductivity by disorder. The result is a nonmonotonic dependence of T_c on doping. The pseudogap phase is described on the basis of the Franz-Millis model of superconducting fluctuations, consisting of small superconducting domains with uncorrelated supercurrents. The calculated characteristics, namely, the spectral function, the inelastic neutron scattering cross section, and the spin susceptibility agree with experimental data.

Inverse Bremsstrahlung absorption of dense fully ionized plasmas in strong laser fields is investigated starting from a quantum kinetic equation with non-Markovian and field dependent collision integrals. First, the dynamically screened Born approximation is considered. For high-frequency laser fields, quantum statistical expressions for the electrical current density and the cycle-averaged electron–ion collision frequency in terms of the Lindhard dielectric function are derived. The expressions are valid for arbitrary field strength assuming the nonrelativistic case. Numerical results are presented to discuss these quantities as a function of the applied laser field and for different plasma parameters. In particular, the influence of non-Maxwellian electron distribution functions is discussed. Furthermore, a generalized expression for the absorption rate is presented to study effects of strong electron–electron and ion–ion correlations, respectively.

In this paper, Green’s function technique is applied to calculate the EOS for weakly coupled plasmas. Especially, the Montroll–Ward approximation is used to cover the density–temperature region where correlations as well as Fermi statistics have to be taken into account. First, low and high density expansions for the thermodynamic functions of two–component plasmas are presented. Then, the EOS in Montroll– Ward approximation is calculated by numerical evaluation. Results for hydrogen and for an electron–positron–plasma are given. We compare our numerical results with recent data from quantum simulations and use our EOS–theory to investigate correlation and quantum effects of the mean kinetic energy.

It has been established that pair collisions generate in nonequilibrium case non-zero two-particle correlations that are non-diagonal in momentum space and give the essential contribution to the current fluctuations of hot electrons. It is shown here that these correlations give also a contribution to the collision integral, i.e., to kinetic properties of nonequilibrium gas. The expression for the electron energy loss rate P via phonons is re-derived in detail from this point of view. The contribution of the non-diagonal part of the nonequilibrium pair correlator to the phonon-electron collision integral and to P is obtained and explicitly calculated in the electron temperature approximation. It is shown that these results may be obtained from stochastic non-linear kinetic equation with Langevin fluctuation force. Such an approach allows to formulate the simple general conditions under that a contribution of two-particle correlations might be essential in kinetics. The contribution obtained does not contain the extra powers of small gas parameter unlike the equilibrium virial decompositions.

Theoretical investigations of the short-time behavior of many-body systems have shown significant deviations from kinetic energy conservation which is typically assumed in conventional kinetic theory^{1 – 4}. We analyze the requirements on the initial state which lead to heating or cooling, respectively. In particular, we investigate the relaxation after changes in the interaction potential. The theoretical results are complemented by quantum kinetics and molecular dynamics simulations. We discuss the possibility to prepare an overcorrelated state and calculate the temperature reduction.

Recent experimental progress in the field of ultracold atoms and ions ^1,2 has lead to an increasing interest in the theoretical understanding of the dynamics of many-body systems at very low temperatures. Under these conditions, the quantum character of the atoms and ions can play a crucial role. We consider situations where the type of spin statistics the particles obey is changed by laser ionization of atoms. We show that after the ionization of fermionic atoms the ion ensemble undergoes a gradual transition from Fermi to Bose statistics. The temporal behavior of several contributions to the total energy on the time scale of correlation build-up is analyzed. Of special interest is the effective exchange energy which reflects the statistics character and the kinetic energy which indicates, depending on the initial state, heating or cooling of the system ³.

The problem of methods to find time-dependent spectra of resonance fluorescence for a medium excited by a short laser pulse is considered. Using the generalized second Born approximation for BBGKY-hierarchy, along with the method of coherent states, we have derived the equations to describe time-dependent scattering of a laser pulse in a two-level medium omitting the rescattering of spontaneous emission. Transient fluorescence spectra with two components representing elastic and spontaneous emission have been obtained. The corrections to the radiative relaxation operator of an atom subjected to a superstrong laser pulse is considered.

Utilizing the method of coherent states within the Hartree-Fock approximation we derived equations to describe elastic (Rayleigh) scattering of resonant laser radiation in a two-level medium. A concentrated ensemble of atoms was assumed to be contained within a volume much smaller then the resonant wavelength to study the impact of a collective behaviour introduced by the collective operator obtained herein. A method to describe and study the time-dependent spectra of scattered radiation has been developed. The presence of asymmetric components shifted by the Rabi frequency and possibility of multimode pattern in the spectrum of scattered radiation is demonstrated for the medium excited by a short detuned laser pulse.

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No abstract received.

Soon, semiconductor devices will utilize very short gate lengths, of order 10-30 nm. These devices are expected to be dominated by quasi-ballistic, quantum transport in the active region. Simulation has developed as a major tool for predictive behavior of new devices, particularly with kinetic transport handled via a multi-particle Monte Carlo approach. Initial quantum effects have been incorporated into such simulations via an effective potential approach with great success, and we will discuss the application to small fully-depleted silicon-on-insulator devices. As devices grow smaller, however, more advanced techniques, such as the non-equilibrium Green’s functions (NEGF) must be utilized, although there are constraints upon these approaches that must be incorporated into modeling of small devices. Nevertheless, approaches which derive from NEGF, such as the Wigner distribution function, can still be implemented via the kinetic Monte Carlo approaches. The application of this approach to the modeling of a resonant tunneling diode also is discussed. The inclusion of both collisional broadening and the intra-collisional field effect into a Monte Carlo simulation of the Wigner function transport is described.

We present an overview of semiconductor device modeling using non-equilibrium Green function techniques. The various efforts and their associated goals, problems, and solutions tend to naturally divide according to the dimensionality, 1D, 2D, or 3D, of the transport which we use to classify and organize the discussion. Our current efforts are largely focused on 2D and 3D modeling. The theory and approach laid out for 1D serves as the basis for 2D and 3D.

We present a sampling of retarded Schrödinger electron Green’s functions in a number of “non-free particle” situations, along with a discussion of several methods for their determination: (I) the Green’s functions of 3D and 2D electrons in a magnetic field are derived using a gauge transformation that depends on both the source (x′) and field (x) points, which explicitly separates the dependence on x+x’ from a translationally invariant part that satisfies a simpler Green’s function equation, made simpler still by invoking the conservation of angular momentum; (II) we exhibit Schwinger’s operator equation-of-motion method¹ in the determination of the Green’s function for a Schrödinger electron in a 2D saddle potential in an arbitrary time-dependent electric field; (III) a determination of the Green’s function for an electron in a 1D superlattice miniband is carried out in the presence of an axial magnetic field (uniform and constant) and a parallel electric field (uniform with arbitrary time dependence), both fields having arbitrary strength. All of these Green’s functions are exhibited in direct time representation in terms of highly tractable elementary functions, which are generators of transcendental eigenfunctions (for example, Landau states and miniband Wannier-Stark-ladder Bloch states or Houston functions with accelerated Bloch states). The results are representative of some more complex cases we have treated in the literature²⁻¹⁰.

We present a nonequilibrium Green’s function formulation of many-body quantum transport theory for multi-band semiconductor systems with a phonon bath. The equations are expressed exactly in terms of single particle nonequilibrium Green’s functions and self-energies, treating the open electron-hole system in weak interaction with the bath. A decoupling technique is employed to separate the individual band Green’s function equations of motion from one another, with the band-band interaction effects embedded in “cross-band” self-energies. This nonequilibrium Green’s function formulation of quantum transport theory is amenable to solution by parallel computing because of its formal decoupling with respect to interband interactions. Moreover, this formulation also permits coding the simulator of an n-band semiconductor in terms of that for an (n – 1)-band system, in step with the current tendency and development of programming technology. Finally, the focus of these equations on individual bands provides a relatively direct route for the determination of carrier motion in energy bands, and to delineate the influence of intra- and inter-band interactions. A detailed description is provided for three-band semiconductor systems.

A generalized Landauer formula, derived with the methods due to Keldysh, and Baym and Kadanoff, is gaining widespread use in the modeling of transport in a large number of different mesoscopic systems. We review some of the recent developments, including transport in semiconductor superlattices, calculation of noise, and nanoelectromechanical systems.

The hydrodynamic picture represents the properties of quantum states as a set of real fields, the simplest of which are the current density and carrier probability density. The topology of the velocity flow provides a useful insight into transport in inhomogeneous systems such as coherent transport through open quantum dots. A formalism is derived which replaces the standard Schrödinger equation with a self-consistent set of coupled transport equations including a vector quantum potential which subsumes effects of vortex formation. Quantised vortex flows are found to be ubiquitous and conditions for their formation are derived. Bohm trajectories are equivalent to streamlines in steady flows but otherwise correspond to trajectories of a particle evolving classically under the actual potential plus the quantum potential. The hydrodynamic and Bohm pictures are briefly reviewed and their correspondence with non-equilibrium Green function and Wigner function theory is critically examined. The concepts are illustrated by application to quantum transport in strong electric fields and to transport in the presence of non-self-averaging atomistic disorder.

We propose a generalization of the nonequilibrium Green’s functions to treatment of open systems, to account for the influence of the system-environment coupling on transport in nonequilibrium mesoscopic systems. Our approach is based on the partial-trace-free time-convolutionless equation of motion for the open system’s reduced density matrix. We generalize the two-time correlation functions, and analyze the behavior of systems in the transient regime, as well as in a far-from-equilibrium steady state.

We calculate the impact ionization rate and the high field electron transport in wide band gap semiconductors and present explicit results for GaN. The band structure is usually considered within the empirical pseudopotential method. Ensemble Monte Carlo simulations of high field electron transport are performed from which the electron distribution function and the drift velocity are derived. The intra-collisional field effect has only minor influence on measurable quantities such as the ionization coefficient so that the band structure itself is important for the results. Therefore, we have performed electron structure calculations using density functional theory in the exact exchange local density approximation. These ab initio band structures are more reliable than the rather approximate results of the empirical pseudopotential method.

The induced magnetic moment of a semiconductor tunnel-coupled double quantum wire system is analyzed in the case when the parallel wires are connected to leads in a series arrangement, with their length perpendicular to the lead-to-lead current. We derive and solve the equations of motion for the double-wire electron Green’s functions using the transfer-tunneling Hamiltonian formalism. This generalizes the Meir-Wingreen approach to structures having a continuous spectrum. The solution is employed to determine the average magnetic moment of the double-wire system induced by a magnetic field applied perpendicular to the plane of the wires and leads. We show that, at low temperature, the magnetic response of such a system can be either diamagnetic or paramagnetic depending on the applied bias and the equilibrium chemical potential of the leads. These properties of the double-wire structure introduce a mechanism for the lead-to-lead bias control of the induced magnetic moment of this system.

We present a theory accounting for the origin of high frequency current oscillations in a double barrier quantum well system. The origin of such current oscillations is traced to the development of a dynamic emitter quantum well and the concomitant coupling of the energy levels in the double barrier quantum well system. The relationship between the oscillation frequency and the energy level structure of the system is expressed as ν = ΔE₀/ħ: A self-consistent, time-dependent Wigner-Poisson numerical computer experiment is used to exhibit remarkable intrinsic, sustained current oscillations in the double-barrier quantum well at terahertz frequencies; and a procedure for calculating ΔE₀, the energy difference at time t₀ (defined such that the contribution to the energy difference from the potential oscillation is zero) is also presented. The simulated oscillation frequency determined using the Wigner-Poisson analysis is in very good agreement with that calculated using a Schrödinger equation with a self-consistent potential determined from the Poisson equation.

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Light emission of semiconductors is discussed for stationary excitation of arbitrary strength ranging from the low density and purely excitonic state up to the fully ionized state of the excited electron-hole plasma. Footing on nonequilibrium Green’s functions the quantized electromagnetic field is consistently considered inside the semiconductor and for the surrounding vacuum outside solving the Dyson equation of photons. In contrast to common understanding it appears, that light emission and even lasing goes always spontaneously, however into a photonic density of states which is strongly renormalized compared with the vacuum one due to the presence of the semiconductor gain medium. The many-body effects of the electron-hole plasma are accounted for within a T-matrix approach for quasiequilibrium, i.e. if optical losses in the carrier kinetics can be neglected. Emphasis is laid on a consistent treatment of bound and ionized electron-hole pair states and the correct description of the crossover from absorption to gain at the chemical potential. Basing on this analysis apparently different phenomena as Bose-Einstein condensation, excitonic gain or lasing, and the Mott-transition will be discussed and their intimate connection qualitatively demonstrated.

The two-particle Green’s function is analyzed with respect to both static and dynamic interaction. Care is taken to correctly account for the effect of phase space filling, which suggests the result that two-particle complexes obey rather Fermi-Dirac like statistics despite of their boson-like commutator relations.

Coulomb correlations in electron-phonon scattering processes have an important influence on linear and nonlinear optical spectra of semiconductors. They are not only of quantitative importance with respect to renormalization and dephasing of electron-hole pairs, but lead also to qualitative new features, which are missing if the final states of a scattering process are uncorrelated. We review a Green’s function theory, which takes these effects into account in a systematic manner within a selfconsistent one-phonon approximation. The theory is formulated in the time domain and allows for practicable calculations within the two-time formulation of quantum kinetics. We also discuss the reduction of the theory to quantum kinetics in terms of single-time functions by means of a generalized Kadanoff-Baym ansatz and the relationship to the density matrix approach. First numerical studies demonstrate the importance of final-state Coulomb correlations even in the linear case for II-IV bulk semiconductors and quantum-well structures.

We investigate the influence of quantum kinetic effects connected with the scattering of carriers with plasmons and LO-phonons on the excitonic absorption in ZnSe within a linear response theory. Within our approach the quantum–kinetic effects are reflected in a dependence of the many–body effects on the energy. In contrast to earlier treatments we use a unified description of the longitudinal dielectric function, including coupled plasmon and LO-phonon modes.

Semiconductor quantum wells offer unique and exciting possibilities for the investigation of quantum correlations and quantum coherences. The main focus of this contribution is on two-exciton correlations. On the basis of a T-matrix analysis inolving an excitonic T-matrix we will elucidate several nonlinear optical effects observed in recent experiments. After a brief discussion of the theoretical foundation, including the interface between conventional Green’s functions techniques and the dynamics-controlled truncation formalism (a generalized density matrix approach), we discuss the excitonic T-matrix analysis of excitons in two dimensions and the ramifications of the spatial dimension being two in the case of quantum wells. The application and verification of the excitonic T-matrix description is done for a semiconductor quantum well inside an optical microcavity. In particular, we discuss evidence of two-exciton continuum correlations that cannot be understood within the second Born approximation.

The linear absorption of a two-band semiconductor with Coulomb interaction and intermediate electron-LO-phonon coupling is considered. Special attention is payed to the polaron renormalization of the carrier spectrum. It is shown that this may play an important role in the appearance of phonon sidebands in vertexless calculations. To this end the generalized Kadanoff–Baym ansatz (GKBA) is compared with an alternative ansatz which does not rely on the factorization of the spectral Green’s function.

In mixed semiconductor crystals, the alloy disorder in the valence and conduction bands is statistically correlated. This leads to kinematical correlations in the motion of electrons and holes analogous to exciton effects. This mechanism, known to affect the linear optical response, is presently shown to act also in the non-linear regime induced by strong short light pulses. A direct numerical solution of the Kadanoff–Baym equations for non-equilibrium Green functions (with a self-consistent and conserving single-site approximation for the self-energy) is used to study correlation in the electron and hole photoexcited populations, and also an additional transient light-induced band hybridization which modifies the full optical vertex.

We apply the exact diagonalization technique to calculate the ground and excited states of a bipolar artificial molecule composed of two vertically coupled quantum dots containing different types of carriers – electrons and holes – in equilibrium. In this system, the magnetic field tunes the relative role of intra-dot Coulomb interaction while the inter-dot separation sets the strength of inter-dot correlations. We find an intricate pattern of the switching of the ground-state angular momentum with increasing magnetic field and a rearrangement of approximate single-particle levels as a function of the inter-dot coupling strength.

Intershell rotation in small 2D Coulomb clusters in a harmonic confinement potential has been investigated with classical Molecular Dynamics simulations. Exciting rotation of the outer shell by application of a short external force, three different methods were used to determine the intershell barrier: a) from the work of the external force; b) from the fluctuation of angular kinetic energy; c) from the change of the potential energy of the cluster. Comparing the results to independent Monte Carlo calculations allows us to conclude that methods b) and c) are best suited for a dynamic calculation of intershell rotation barriers.

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Within finite temperature field theory, we show that truncated non-perturbative self-consistent Dyson resummation schemes can be renormalized with local vacuum counterterms. For this the theory has to be renormalizable in the usual sense and the self-consistent scheme must follow Baym’s Φ-derivable concept. Our Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization scheme leads to renormalized self-consistent equations of motion. At the same time the corresponding two-particle irreducible (2PI) generating functional and the thermodynamic potential can be renormalized with the same counterterms used for the equations of motion. This guarantees the standard Φ-derivable properties like thermodynamic consistency and exact conservation laws also for the renormalized approximation schemes. We also give a short overview over symmetry properties of the various functions defined within the 2PI scheme for the case that the underlying classical field theory has a global linearly realized symmetry.

We consider the nonequilibrium evolution of an O(N)–symmetric scalar quantum field theory using a systematic two–particle irreducible 1/N–expansion to next-to-leading order, which includes scattering and memory effects. The corresponding “full Kadanoff-Baym equations” are solved numerically without further approximations. This allows one to obtain a controlled nonperturbative description of far-from-equilibrium dynamics and the late-time approach to quantum thermal equilibrium. Employing, in addition, a first-order gradient expansion for the Wigner transformed correlators we derive kinetic equations. In contrast to standard descriptions based on loop expansions, our equations remain valid for nonperturbatively large fluctuations. As an application, we discuss the fluctuation dominated regime following parametric resonance in quantum field theory.

In the framework of the Green’s function formalism at finite temperatures super-fluidity of nuclear matter with neutron–proton (np) pairing correlations is studied. It is shown that at low densities the equations for the energy gap in the spectrum of quasiparticles and chemical potential allow for solutions with negative chemical potential, that corresponds to appearance of Bose–Einstein condensation (BEC) of deuterons in the low density region of nuclear matter. Interaction between nucleons is described by the effective zero range force, developed to reproduce the energy gap in the isospin singlet pairing channel, obtained with the use of the Paris NN potential. The effects of isospin asymmetry are also discussed.

A kinetic approach to scalar QED in a strong and time-dependent electric field is studied taking into account self-interactions. A quantum kinetic equation, including both a term describing crossed effects of vacuum particle creation and self-interaction in the Hartree-Fock approximation and a collision integral of Boltzmann type, is derived. The source term and collision integrals are characterized by a non-Markovian structure pointing to a complicated interplay of time scales in any realistic scenario.

We study particle–antiparticle pair production under action of a strong time dependent spatially homogeneous electric field in the presence of a collinear constant magnetic field. We derive the kinetic equation for such a field configuration for fermions and bosons in the framework of the Schwinger mechanism of vacuum tunneling. We show the enhancement of pair production for fermions (suppression for bosons) with the increase of the magnetic field as in the case of a constant electromagnetic field. We have constructed a closed set of equations, which can be applied to some actual problems with manifestation of strong electromagnetic fields, e.g., it is essential in the framework of the Flux Tube Model of Quark–Gluon Plasma generation; for describing some cosmological objects and especially because of the planned experiments on creation of subcritical fields in X-Free Electron Laser pulses.

A Vlasov type quantum kinetic equation for the quark subsystem in a strong quasi-classical gluon field is derived in the covariant single-time formalism. As a result of spinor and color decompositions, the system of equations for the corresponding components of the Wigner function is obtained. A few particular cases are discussed in detail.

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We give a short overview of time-dependent density functional theory applied to dynamics of metal clusters. The actual level of treatment is the time-dependent local-density approximation (TDLDA) coupled to classical molecular dynamics for the ions (TDLDA-MD). We discuss the performance of LDA for this particular application and necessary extensions, as e.g. the self-interaction correction (SIC). Finally, a few selected examples of cluster dynamics are presented, from optical response to pump and probe experiments.

We give an overview of the underlying concepts of time-dependent current-density-functional theory (TDCDFT). We show how the basic equations of TDCDFT can be elegantly derived using the time contour method of nonequilibrium Green function theory. We further demonstrate how the formalism can be used to derive explicit equations for the exchange-correlation vector potentials and integral kernels for the Kohn-Sham equations and their linearized form.

We present first-principle path integral Monte-Carlo (PIMC) studies of strongly correlated electron-hole complexes such as excitons, trions (charged excitons) and biexcitons in Al_xGa_1–xAs quantum-well structures. The correlation and binding energies are calculated as function of quantum well width and compared with available experimental^1,2,3,4 and theoretical^5,6 data. The only approximation made is the separation of particle motion parallel and perpendicular to the quantum well plane. The obtained very accurate PIMC results for experimentally measured binding energies validates the applicability of our approximation in a wide range of quantum well widths 10Å ≤ L ≤ 250Å. As in the experiments, we observe a maximum of the binding energies in GaAs/AlGaAs quantum well samples around L = 40Å. We clearly show that the physical reason is non-monotonic dependence of the electron (hole) confinement on the well width. The developed method is a powerful tool for further investigation of temperature and many-body effects on bound states in heterostructures (e.g. dependence on finite exciton, biexciton densities or disorder).

Strengths and limitations of quantum kinetic simulations of correlated quantum Coulomb systems are reviewed. To make accurate simulations in the regime of strong coupling possible it is proposed to use first-principle simulation data as an input. A new approach is developed which allows for a first principle computation of dielectric and dynamic properties of quantum plasmas including situations of strong coupling, partial ionization/dissociation and Wigner crystallization.

Recent developments of the application of the Wigner-path (WP) method to nonequilibrium quantum transport in mesoscopic systems are presented. The concept of WP allows the formulation of a Monte Carlo simulation which is quantum mechanically rigorous and yet very similar to the one used in semiclassical transport theory. Scatterings with the potential profile and with phonons are included in the path in a way that takes automatically into account all quantum effects, such as intracollisional field effect and collisional broadening.

The nonstationary tunneling of two interacting identical particles is analyzed. The study is performed by means of computer simulation with the method of quantum molecular dynamics (QMD). It is based on the Wigner representation of quantum mechanics and in the framework of this method ensembles of semiclassical trajectories are used to solve the quantum Wigner-Liouville equation. We have developed the QMD method to take into account the influence of exchange and interaction between particles. The role of direct and exchange interactions in tunneling is analyzed.

A fast and efficient numerical-analytical approach is proposed for modeling complex behaviour in the BBGKY–hierarchy of kinetic equations. Our calculations are based on variational and multiresolution approaches in the basis of polynomial tensor algebras of generalized coherent states/wavelets. We construct the representation for the hierarchy of reduced distribution functions via multiscale decomposition in highly-localized eigenmodes. Numerical modeling shows the creation of various internal structures from localized modes, which are related to localized or chaotic type of behaviour and the corresponding pattern (waveleton) formation.

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Name Index.

Keyword Index.

Workshop Photos.